Apparatus for controlling air-fuel ratio of internal combustion engine

ABSTRACT

The air-fuel ratio of an air-fuel mixture to be combusted in an internal combustion engine  1  is manipulated to converge an output VO 2 /out of an O 2  sensor  6  which is disposed downstream of a catalytic converter  3  to a target value VO 2 /TARGET depending on an operating state of the internal combustion engine  1 , while at the same time a deterioration evaluating parameter is determined from time-series data of the output VO 2 /out of the O 2  sensor  6 . The value of the deterioration evaluating parameter is corrected depending on an average value of the output VO 2 /out of the O 2  sensor  6  or the target value VO 2 /TARGET, and the deteriorated state of the catalytic converter  3  is evaluated based on the corrected value of the deterioration evaluating parameter.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an apparatus for controlling theair-fuel ratio of an internal combustion engine, and more particularlyto an air-fuel ratio control apparatus which is capable of evaluating adeteriorated state of a catalytic converter for purifying an exhaustgas.

2. Description of the Related Art

There have been known various techniques for controlling the air-fuelratio of an air-fuel mixture to be combusted by an internal combustionengine in order to achieve an appropriate purifying capability of acatalytic converter for purifying exhaust gases, such as a three-waycatalytic converter, disposed in the exhaust passage of the internalcombustion engine. See, for example, Japanese laid-open patentpublication No. 9-324681 or U.S. Pat. No. 5,852,930, and Japaneselaid-open patent publication No. 11-153051 or U.S. Pat. No. 6,112,517.According to the disclosed systems, an O₂ sensor (oxygen concentrationsensor) is disposed downstream of the catalytic converter, and theair-fuel ratio of an air-fuel mixture is controlled according to afeedback control process (specifically, a sliding mode control process)in order to converge the output of the O₂ sensor to a predeterminedtarget value (constant value).

The catalytic converter is progressively deteriorated and its purifyingcapability is lowered as the internal combustion engine is used over along period of time. Therefore, the catalytic converter needs to bereplaced with a brand-new catalytic converter when it has beendeteriorated to a certain extent. It has thus been desired in the art tobe able to adequately evaluate a deteriorated state of a catalyticconverter which is combined with an internal combustion engine. Knownsystems for evaluating a deteriorated state of a catalytic converter aredisclosed in Japanese patent No. 2526640 and Japanese laid-open patentpublication No. 7-19033, for example. According to these disclosedevaluating systems, when the air-fuel ratio of an air-fuel mixture to becombusted by an internal combustion engine is changed from a leanervalue to a richer value or from a richer value to a leaner value, thetime required until the output of an O₂ sensor that is positioneddownstream of the catalytic converter is inverted and the period inwhich the output of the O₂ sensor is inserted are measured. Thedeteriorated state of the catalytic converter is evaluated based on theevaluated values.

The disclosed systems, however, make it difficult to achieve anappropriate purifying capability of the catalytic converter at the timeits deteriorated state is evaluated because it is necessary topositively change the air-fuel ratio in order to evaluate thedeteriorated state of the catalytic converter. Stated otherwise, it isdifficult for the disclosed system to evaluate the deteriorated state ofthe catalytic converter while at the same time maintaining anappropriate purifying capability of the catalytic converter.

The applicant of the present application has made an effort to developan apparatus capable of evaluating a deteriorated state of a catalyticconverter while controlling an air-fuel ratio to converge the output ofan O₂ sensor that is positioned downstream of the catalytic converter toa given target value, i.e., maintaining a good purifying capability ofthe catalytic converter. For example, reference should be made toJapanese patent application No. 2000-139860 (Japanese laid-open patentpublication No. 2000-241349). The disclosed system is based on the factthat the value of a certain parameter which is determined from the dataof the output of an O₂ sensor that is positioned downstream of thecatalytic converter exhibits characteristic properties as thedeterioration of the catalytic converter progresses, and evaluates thedeteriorated state of the catalytic converter based on the value of theparameter. The system thus arranged makes it possible to evaluate thedeteriorated state of the catalytic converter while maintaining a goodpurifying capability of the catalytic converter.

In the above arrangements for controlling the air-fuel ratio to convergethe output of the O₂ sensor that is positioned downstream of thecatalytic converter to a given target value (see Japanese laid-openpatent publication No. 9-324681 or U.S. Pat. No. 5,852,930, and Japaneselaid-open patent publication No. 11-153051 or U.S. Pat. No. 6,112,517),the target value for the output of the O₂ sensor positioned downstreamof the catalytic converter is basically a certain constant value.However, depending on the type of the catalytic converter, it mayoccasionally preferable to make variable the target value for the outputof the O₂ sensor depending on the operating state of the internalcombustion engine for the purpose of enabling the catalytic converter toachieve an appropriate purifying capability.

However, a control system for controlling the air-fuel ratio accordingto the variable target value for the output of the O₂ sensor positioneddownstream of the catalytic converter tends to suffer certain drawbacks,given below, with respect to the reliability of the evaluation of thedeteriorate state of the catalytic converter at the time of evaluatingthe deteriorate state of the catalytic converter while at the same timecontrolling the air-fuel ratio.

Since the output of the O₂ sensor is nonlinear with respect to theoxygen concentration (see the solid-line curve a in FIG. 2 of theaccompanying drawings), if the target value for the output of the O₂sensor changes, then the output of the O₂ sensor changes differentlywhen it is converged to the target value. As a result, the value of theparameter that is uniquely determined from the data of the output of theO₂ sensor is affected by a factor other than the deteriorate state ofthe catalytic converter, and the deteriorate state of the catalyticconverter which is evaluated based on the value of the parameter tendsto become less reliable than expected.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide anapparatus for controlling the air-fuel ratio of an internal combustionengine to converge the output of an O₂ sensor that is positioneddownstream of a catalytic converter to a target value which is variablyestablished depending on the operating state of the internal combustionengine, the apparatus being capable of appropriately evaluating thedeteriorated state of the catalytic converter while compensating foradverse effects which are produced by making the target value variable.

To achieve the above object, there is provided in accordance with afirst aspect of the present invention an apparatus for controlling theair-fuel ratio of an internal combustion engine, comprising an oxygenconcentration sensor disposed downstream of a catalytic converter whichis disposed in an exhaust passage of an internal combustion engine,air-fuel ratio manipulating means for manipulating the air-fuel ratio ofan air-fuel mixture to be combusted in the internal combustion engine toconverge an output of the oxygen concentration sensor to a target valueestablished depending on an operating state of the internal combustionengine, parameter generating means for generating a deteriorationevaluating parameter for evaluating a deteriorated state of thecatalytic converter according to an algorithm predetermined from data ofthe output of the oxygen concentration sensor while the air-fuel ratiomanipulating means is manipulating the air-fuel ratio of the air-fuelmixture, and deteriorated state evaluating means for correcting thedeterioration evaluating parameter depending on an average value of thetarget value or the output of the oxygen concentration sensor, andevaluating the deteriorated state of the catalytic converter based onthe corrected deterioration evaluating parameter.

According to a second aspect of the present invention, there is alsoprovided an apparatus for controlling the air-fuel ratio of an internalcombustion engine, comprising an oxygen concentration sensor disposeddownstream of a catalytic converter which is disposed in an exhaustpassage of an internal combustion engine, air-fuel ratio manipulatingmeans for manipulating the air-fuel ratio of an air-fuel mixture to becombusted in the internal combustion engine to converge an output of theoxygen concentration sensor to a target value established depending onan operating state of the internal combustion engine, parametergenerating means for generating a deterioration evaluating parameter forevaluating a deteriorated state of the catalytic converter according toan algorithm predetermined from data of the output of the oxygenconcentration sensor while the air-fuel ratio manipulating means ismanipulating the air-fuel ratio of the air-fuel mixture, anddeteriorated state evaluating means for evaluating the deterioratedstate of the catalytic converter by comparing the deteriorationevaluating parameter with a predetermined decision value, thedeteriorated state evaluating means comprising means for establishingthe decision value depending on an average value of the target value orthe output of the oxygen concentration sensor, and means for evaluatingthe deteriorated state of the catalytic converter by comparing theestablished decision value with the deterioration evaluating parameter.

With the above arrangement, the deterioration evaluating parametergenerated from the data of the output of the oxygen concentration sensoris corrected according to the first aspect or the decision value to becompared with the deterioration evaluating parameter is establishedaccording to the second aspect, depending on the average value of thetarget value or the output of the oxygen concentration sensor, or statedotherwise depending on what level the output of the oxygen concentrationsensor is controlled at by the air-fuel ratio manipulating means. Inthis manner, even if the target value for the output of the oxygenconcentration sensor is established variably depending on the operatingstate of the internal combustion engine, the deteriorated state of thecatalytic converter can be evaluated irrespectively of the target value.

Specifically, according to the first aspect of the present invention,the deterioration evaluating parameter is corrected depending on theaverage value of the target value or the output of the oxygenconcentration sensor, so that the corrected deterioration evaluatingparameter varies depending on only the deteriorated state of thecatalytic converter, not the target value for the output of the oxygenconcentration sensor. As a result, the deteriorated state of thecatalytic converter can be evaluated appropriately based on thecorrected deterioration evaluating parameter. According to the secondaspect of the present invention, though the deterioration evaluatingparameter is affected by the target value for the output of the oxygenconcentration sensor, the decision value to be compared with thedeterioration evaluating parameter is established depending on theaverage value of the target value or the output of the oxygenconcentration sensor for thereby providing the same advantages as thoseof the first aspect. According to the present invention, because thedeterioration evaluating parameter is generated from the data of theoutput of the oxygen concentration sensor while the air-fuel ratiomanipulating means is manipulating the air-fuel ratio of the air-fuelmixture, i.e., the output of the oxygen concentration sensor is beingconverged to the target value, the deteriorated state of the catalyticconverter can be evaluated highly reliably while the desired purifyingperformance of the catalytic converter is well maintained.

In the above apparatus, the deterioration evaluating parameterpreferably comprises data representing a variation of a deteriorationevaluating linear function whose variable components are represented bytime-series data of the output of the oxygen concentration sensor.

Specifically, while the output of the oxygen concentration sensor isbeing converged to the target value, when the value of a suitable linearfunction, i.e., a function represented by a linear combination oftime-series data of the output of the oxygen concentration sensor, whosevariable components are represented by time-series data of the output ofthe oxygen concentration sensor, is determined from the time-series dataof the output of the oxygen concentration sensor, the value of thelinear function tends to be characteristically correlated to the degreeof process of the deterioration of the catalytic converter. For example,when the catalytic converter is substantially brand-new, the value ofthe linear function is collected in the vicinity of a certain value. Asthe deterioration of the catalytic converter progresses, the value ofthe linear function tends to be spaced from the certain value.Therefore, the value of the linear function varies more greatly as thedeterioration of the catalytic converter progresses.

By generating data representing the variation of the value of the linearfunction as the deterioration evaluating parameter, the correlationbetween the value of the deterioration evaluating parameter and thedeteriorated state of the catalytic converter is increased, and hencethe reliability of the evaluation of the deteriorated state of thecatalytic converter is increased.

Though the deterioration evaluating parameter may be represented by thesquare of the difference between the value of the deteriorationevaluating function and a certain value or the absolute value of thedifference, the parameter generating means preferably comprises meansfor generating the deterioration evaluating parameter by effectinglow-pass filtering on the square value or absolute value of thedifference between values of the time-series data of the output of theoxygen concentration sensor and a predetermined value as a central valueof the deterioration evaluating linear function.

The deterioration evaluating parameter determined by effecting low-passfiltering on the square value or absolute value of the difference is ofan appropriate value representing the variation of the value of thedeterioration evaluating linear function, and increases monotonously asthe deterioration of the catalytic converter progresses. Therefore, thedeteriorated state of the catalytic converter can be evaluated highlyreliably based on the value of the deterioration evaluating parameter.

With the deterioration evaluating linear function being used todetermine the deterioration evaluating parameter, the air-fuel ratiomanipulating means preferably comprises means for sequentiallygenerating a manipulated variable for manipulating the air-fuel ratioaccording to a sliding mode control process to converge the output ofthe oxygen concentration sensor to the target value, and manipulatingthe air-fuel ratio depending on the manipulated variable, thedeterioration evaluating linear function comprising a linear functiondetermined depending on a switching function used in the sliding modecontrol process.

Specifically, the above tendency of the value of the deteriorationevaluating linear function depending on the deteriorated state of thecatalytic converter is likely to manifest itself when the manipulatedvariable (e.g., a target air-fuel ratio) for manipulating the air-fuelratio with the air-fuel ratio manipulating means is generated accordingto a sliding mode control process which is a feedback control process.In the case where the air-fuel ratio is manipulated according to thesliding mode control process, the deterioration evaluating linearfunction highly correlated to the deteriorated state of the catalyticconverter is closely related to a switching function used in the slidingmode control process, and it is preferable to use a linear functiondetermined depending on the switching function as the deteriorationevaluating linear function.

More specifically, in the sliding mode control process, the linearfunction whose variable components are represented by time-series dataof the difference between the output of the oxygen concentration sensorand the target value is used as the switching function. In the casewhere this switching function is used in the sliding mode controlprocess, the deterioration evaluating linear function should preferablycomprise a linear function where coefficient values of its variablecomponents are identical to coefficient values of variable components ofthe switching function. The linear function may be the switchingfunction itself which is used in the sliding mode control process.

By thus using the linear function determined depending on the switchingfunction for the sliding mode control process as the deteriorationevaluating linear function, the correlation between the value of thedeterioration evaluating linear function and the deteriorated state ofthe catalytic converter manifests itself, allowing the deterioratedstate of the catalytic converter to be evaluated properly based on thevalue of the deterioration evaluating linear function.

The above and other objects, features, and advantages of the presentinvention will become apparent from the following description when takenin conjunction with the accompanying drawings which illustrate preferredembodiments of the present invention by way of example.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an overall system arrangement of anapparatus for controlling the air-fuel ratio of an internal combustionengine according to a first embodiment of the present invention;

FIG. 2 is a diagram showing output characteristics of an O₂ sensor andan air-fuel ratio sensor used in the apparatus shown in FIG. 1;

FIG. 3 is a block diagram showing a basic arrangement of a majorcomponent of the apparatus shown in FIG. 1;

FIG. 4 is a diagram illustrative of a sliding mode control processcarried out in the apparatus shown in FIG. 1;

FIG. 5 is a diagram illustrative of a process of evaluating thedeteriorated state of a catalytic converter employed by the apparatusshown in FIG. 1;

FIG. 6 is a diagram illustrative of the process of evaluating thedeteriorated state of the catalytic converter employed by the apparatusshown in FIG. 1;

FIG. 7 is a diagram illustrative of the process of evaluating thedeteriorated state of the catalytic converter employed by the apparatusshown in FIG. 1;

FIG. 8 is a diagram illustrative of the process of evaluating thedeteriorated state of the catalytic converter employed by the apparatusshown in FIG. 1;

FIG. 9 is a diagram illustrative of the process of evaluating thedeteriorated state of the catalytic converter employed by the apparatusshown in FIG. 1;

FIG. 10 is a block diagram of an adaptive controller employed in theapparatus shown in FIG. 1;

FIG. 11 is a flowchart of a main routine of an engine-side control unitof the apparatus shown in FIG. 1;

FIG. 12 is a flowchart of a subroutine of the main routine shown in FIG.11;

FIG. 13 is a flowchart of a main routine of an exhaust-side control unitof the apparatus shown in FIG. 1;

FIG. 14 is a flowchart of a subroutine of the main routine shown in FIG.13;

FIG. 15 is a flowchart of a subroutine of the main routine shown in FIG.13;

FIG. 16 is a flowchart of a subroutine of the main routine shown in FIG.13;

FIG. 17 is a flowchart of a subroutine of the main routine shown in FIG.13;

FIG. 18 is a flowchart of a subroutine of the subroutine shown in FIG.17

FIG. 19 is a flowchart of a subroutine of the subroutine shown in FIG.17

FIG. 20 is a flowchart of a subroutine of the subroutine shown in FIG.17

FIG. 21 is a flowchart of a processing sequence of an apparatus forcontrolling the air-fuel ratio of an internal combustion engineaccording to a second embodiment of the present invention;

FIG. 22 is a flowchart of a subroutine of the processing sequence shownin FIG. 21; and

FIG. 23 is a diagram showing a data table used in the subroutine shownin FIG. 22.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An apparatus for controlling the air-fuel ratio of an internalcombustion engine according to a first embodiment of the presentinvention will be described below with reference to FIGS. 1 through 20.

FIG. 1 shows in block form the apparatus for controlling the air-fuelratio of an internal combustion engine according to the first embodimentof the present invention. As shown in FIG. 1, an internal combustionengine 1 such as a four-cylinder internal combustion engine is mountedas a propulsion source on an automobile or a hybrid vehicle, forexample. When a mixture of fuel and air is combusted in each cylinder ofthe internal combustion engine 1, an exhaust gas is generated andemitted from each cylinder into a common exhaust pipe (exhaust passage)2 positioned near the internal combustion engine 1, from which theexhaust gas is discharged into the atmosphere. Two catalytic converters3, 4, each comprising a three-way catalytic converter, are mounted inthe common exhaust pipe 2 at successively downstream locations thereon.

The upstream catalytic converter 3 is evaluated for its deterioratedstate according to the present embodiment. The downstream catalyticconverter 4 may be dispensed with.

The apparatus basically serves to control the air-fuel ratio of anair-fuel mixture combusted by the internal combustion engine 1, in orderto achieve an optimum purifying capability of the catalytic converter 3.While controlling the air-fuel ratio, the apparatus also evaluates thedeteriorated state of the catalytic converter 3.

In order to perform the above processing, the apparatus has an air-fuelratio sensor 5 mounted on the exhaust pipe 2 upstream of the catalyticconverter 3, or more precisely at a position where exhaust gases fromthe cylinders of the internal combustion engine 1 are put together, anO₂ sensor (oxygen concentration sensor) 6 mounted on the exhaust pipe 2downstream of the catalytic converter 3 and upstream of the catalyticconverter 4, and a control unit 7 for carrying out a control process(described later on) and evaluating the deteriorated state of thecatalytic converter 3 based on detected outputs from the sensors 5, 6.

The control unit 7 is supplied with detected outputs from the sensors 5,6 and also detected outputs from various other sensors (not shown) fordetecting an operating state of the internal combustion engine 1,including a engine speed sensor, an intake pressure sensor, a coolanttemperature sensor, etc.

The O₂ sensor 6 comprises an ordinary O₂ sensor for generating an outputVO2/OUT having a level depending on the oxygen concentration in theexhaust gas that has passed through the catalytic converter 3, i.e., anoutput VO2/OUT representing a detected value of the oxygen concentrationof the exhaust gas. The oxygen concentration in the exhaust gas iscommensurate with the air-fuel ratio of an air-fuel mixture which, whencombusted, produces the exhaust gas. The output VO2/OUT from the O₂sensor 6 will change with high sensitivity in proportion to the oxygenconcentration in the exhaust gas, with the air-fuel ratio correspondingto the oxygen concentration in the exhaust gas being in a range Δ closeto a stoichiometric air-fuel ratio, as indicated by the solid-line curvea in FIG. 2. At oxygen concentrations corresponding to air-fuel ratiosoutside of the range Δ, the output VO2/OUT from the O₂ sensor 6 issaturated and is of a substantially constant level. The sensitivity ofthe output VO2/OUT from the O₂ sensor 6 with respect to the oxygenconcentration, i.e., the rate of change of the output VO2/OUT withrespect to a change of the oxygen concentration, is maximum in thevicinity of a central level VO2/OUT1 intermediate between upper andlower limit levels of the output VO2/OUT, and is slightly lower atlevels of the output VO2/OUT which are higher and lower than the centrallevel VO2/OUT1, e.g., at a level VO2/OUT2 and a level VO2/OUT3.

The air-fuel ratio sensor 5 generates an output KACT representing adetected value of the air-fuel ratio (which may hereinafter referred tosimply as “air-fuel ratio of the internal combustion engine 1”) which isrecognized from the concentration of oxygen in the exhaust gas thatenters the catalytic converter 3. The air-fuel ratio sensor 5 comprisesa wide-range air-fuel ration sensor disclosed in detail in Japaneselaid-open patent publication No. 4-369471, which corresponds to U.S.Pat. No. 5,391,282. As indicated by the solid-line curve b in FIG. 2,the air-fuel ratio sensor 5 generates an output whose level isproportional to the concentration of oxygen in the exhaust gas in awider range than the O₂ sensor 6. Stated otherwise, the air-fuel ratiosensor 5 (hereinafter referred to as “LAF sensor 5”) generates an outputwhose level corresponds to the concentration of oxygen in the exhaustgas in a wide range of air-fuel ratios.

The control unit 7 comprises a control unit 7 a (hereinafter referred toas an “exhaust-side control unit 7 a”) for performing a process ofcalculating a target air-fuel ratio KCMD as a manipulated variable fordetermining the air-fuel ratio of the internal combustion engine 1, orspecifically a target value for the air-fuel ratio detected by the LAFsensor 5, and a process of evaluating the deteriorated state of thecatalytic converter 3, and a control unit 7 b (hereinafter referred toas an “engine-side control unit 7 b”) for carryout out a process ofcontrolling the air-fuel ratio of the air-fuel mixture combusted by theinternal combustion engine 1 by adjusting the amount of fuel injectedinto (the amount of fuel supplied to) the internal combustion engine 1depending on the target air-fuel ratio KCMD.

The control units 7 a, 7 b comprise a microcomputer, and perform theirrespective control processes in given control cycles. In the presentembodiment, each of the control cycles in which the exhaust-side controlunit 7 a performs its processes of generating the target air-fuel ratioKCMD and evaluating the deteriorated state of the catalytic converter 3has a predetermined period (e.g., ranging from 30 to 100 ms) in view ofthe dead time (described later on) present in the catalytic converter 3,calculating loads, etc.

The process of adjusting the fuel injection quantity, which is carriedout by the engine-side control unit 7 b, needs to be in synchronism withthe rotational speed of the internal combustion engine 1, or morespecifically combustion cycles of the internal combustion engine 1.Therefore, each of the control cycles in which the engine-side controlunit 7 b performs its process has a period in synchronism with thecrankshaft angle period (so-called TDC) of the internal combustionengine 1.

The constant period of the control cycles of the exhaust-side controlunit 7 a is longer than the crankshaft angle period (so-called TDC) ofthe internal combustion engine 1.

The engine-side control unit 7 b will further be described below withreference to FIG. 1. The engine-side control unit 7 b has, as its mainfunctions, a basic fuel injection quantity calculator 8 for determininga basic fuel injection quantity Tim to be injected into the internalcombustion engine 1, a first correction coefficient calculator 9 fordetermining a first correction coefficient KTOTAL to correct the basicfuel injection quantity Tim, and a second correction coefficientcalculator 10 for determining a second correction coefficient KCMDM tocorrect the basic fuel injection quantity Tim.

The basic fuel injection quantity calculator 8 determines a referencefuel injection quantity (fuel supply quantity) from the rotational speedNE and intake pressure PB of the internal combustion engine 1 using apredetermined map, and corrects the determined reference fuel injectionquantity depending on the effective opening area of a throttle valve(not shown) of the internal combustion engine 1, thereby calculating abasic fuel injection quantity Tim.

The first correction coefficient KTOTAL determined by the firstcorrection coefficient calculator 9 serves to correct the basic fuelinjection quantity Tim in view of an exhaust gas recirculation ratio ofthe internal combustion engine 1, i.e., the proportion of an exhaust gascontained in an air-fuel mixture introduced into the internal combustionengine 1, an amount of purged fuel supplied to the internal combustionengine 1 when a canister (not shown) is purged, a coolant temperature,an intake temperature, etc.

The second correction coefficient KCMDM determined by the secondcorrection coefficient calculator 10 serves to correct the basic fuelinjection quantity Tim in view of the charging efficiency of an air-fuelmixture due to the cooling effect of fuel flowing into the internalcombustion engine 1 depending on a target air-fuel ratio KCMD determinedby the exhaust-side control unit 7 a, as described later on.

The engine-side control unit 7 b corrects the basic fuel injectionquantity Tim with the first correction coefficient KTOTAL and the secondcorrection coefficient KCMDM by multiplying the basic fuel injectionquantity Tim by the first correction coefficient KTOTAL and the secondcorrection coefficient KCMDM, thus producing a demand fuel injectionquantity Tcyl for the internal combustion engine 1.

Specific details of processes for calculating the basic fuel injectionquantity Tim, the first correction coefficient KTOTAL, and the secondcorrection coefficient KCMDM are disclosed in detail in Japaneselaid-open patent publication No. 5-79374 and U.S. Pat. No. 5,253,630,and will not be described below.

The engine-side control unit 7 b also has, in addition to the abovefunctions, a feedback controller 14 for feedback-controlling theair-fuel ratio of the air-fuel mixture to be combusted in the internalcombustion engine 1 by adjusting a fuel injection quantity of theinternal combustion engine 1 so as to converge the output KACT of theLAF sensor 5 (the detected air-fuel ratio of the internal combustionengine 1) toward the target air-fuel ratio KCMD which is calculated bythe exhaust-side control unit 7 a.

The feedback controller 14 comprises a general feedback controller 15for feedback-controlling a total fuel injection quantity for all thecylinders of the internal combustion engine 1 and a local feedbackcontroller 16 for feedback-controlling a fuel injection quantity foreach of the cylinders of the internal combustion engine 1.

The general feedback controller 15 sequentially determines a feedbackcorrection coefficient KFB to correct the demand fuel injection quantityTcyl (by multiplying the demand fuel injection quantity Tcyl) so as toconverge the output KACT from the LAF sensor 5 toward the targetair-fuel ratio KCMD.

The general feedback controller 15 comprises a PID controller 17 forgenerating a feedback manipulated variable KLAF as the feedbackcorrection coefficient KFB depending on the difference between theoutput KACT from the LAF sensor 5 and the target air-fuel ratio KCMDaccording to a known PID control process, and an adaptive controller 18(indicated by “STR” in FIG. 1) for adaptively determining a feedbackmanipulated variable KSTR for determining the feed-back correctioncoefficient KFB in view of changes in operating state of the internalcombustion engine 1 or characteristic changes thereof from the outputKACT from the LAF sensor 5 and the target air-fuel ratio KCMD.

In the present embodiment, the feedback manipulated variable KLAFgenerated by the PID controller 17 is of “1” and can be used directly asthe feedback correction coefficient KFB when the output KACT (thedetected air-fuel ratio) from the LAF sensor 5 is equal to the targetair-fuel ratio KCMD. The feedback manipulated variable KSTR generated bythe adaptive controller 18 becomes the target air-fuel ratio KCMD whenthe output KACT from the LAF sensor 5 is equal to the target air-fuelratio KCMD. A feedback manipulated variable kstr (=KSTR/KCMD) which isproduced by dividing the feedback manipulated variable KSTR by thetarget air-fuel ratio KCMD with a divider 19 can be used as the feedbackcorrection coefficient KFB.

The feedback manipulated variable KLAF generated by the PID controller17 and the feedback manipulated variable kstr which is produced bydividing the feedback manipulated variable KSTR from the adaptivecontroller 18 by the target air-fuel ratio KCMD are selected one at atime by a switcher 20. A selected one of the feedback manipulatedvariable KLAF and the feedback manipulated variable KSTR is used as thefeedback correction coefficient KFB. The demand fuel injection quantityTcyl is corrected by being multiplied by the feedback correctioncoefficient KFB. Details of the general feedback controller 15(particularly, the adaptive controller 18) will be described later on.

The local feedback controller 16 comprises an observer 21 for estimatingreal air-fuel ratios #nA/F (n=1, 2, 3, 4) of the respective cylindersfrom the output KACT from the LAF sensor 5, and a plurality of PIDcontrollers 22 (as many as the number of the cylinders) for determiningrespective feedback correction coefficients #nKLAF for fuel injectionquantities for the cylinders from the respective real air-fuel ratios#nA/F estimated by the observer 21 according to a PID control process soas to eliminate variations of the air-fuel ratios of the cylinders.

Briefly stated, the observer 21 estimates a real air-fuel ratio #nA/F ofeach of the cylinders as follows: A system from the internal combustionengine 1 to the LAF sensor 5 (where the exhaust gases from the cylindersare combined) is considered to be a system for generating an air-fuelratio detected by the LAF sensor 5 from a real air-fuel ratio #nA/F ofeach of the cylinders, and is modeled in view of a detection responsedelay (e.g., a time lag of first order) of the LAF sensor 5 and achronological contribution of the air-fuel ratio of each of thecylinders to the air-fuel ratio detected by the LAF sensor 5. Based onthe modeled system, a real air-fuel ratio #nA/F of each of the cylindersis estimated from the output KACT from the LAF sensor 5.

Details of the observer 21 are disclosed in Japanese laid-open patentpublication No. 7-83094 and U.S. Pat. No. 5,531,208, and will not bedescribed below.

Each of the PID controllers 22 of the local feedback controller 16divides the output signal KACT from the LAF sensor 5 by an average valueof the feedback correction coefficients #nKLAF determined by therespective PID controllers 22 in a preceding control cycle to produce aquotient value, and uses the quotient value as a target air-fuel ratiofor the corresponding cylinder. Each of the PID controllers 22 thendetermines a feedback correction coefficient #nKLAF in a present controlcycle so as to eliminate any difference between the target air-fuelratio and the corresponding real air-fuel ratio #nA/F determined by theobserver 21.

The local feedback controller 16 multiplies a value, which has beenproduced by multiplying the demand fuel injection quantity Tcyl by theselected feedback correction coefficient KFB produced by the generalfeedback controller 15, by the feedback correction coefficient #nKLAFfor each of the cylinders, thereby determining an output fuel injectionquantity #nTout (n=1, 2, 3, 4) for each of the cylinders.

The output fuel injection quantity #nTout thus determined for each ofthe cylinders is corrected for accumulated fuel particles on intake pipewalls of the internal combustion engine 1 by a fuel accumulationcorrector 23 in the engine-side control unit 7 b. The corrected outputfuel injection quantity #nTout is applied to each of fuel injectors (notshown) of the internal combustion engine 1, which injects fuel into eachof the cylinders with the corrected output fuel injection quantity#nTout.

The correction of the output fuel injection quantity in view ofaccumulated fuel particles on intake pipe walls is disclosed in detailin Japanese laid-open patent publication No. 8-21273 and U.S. Pat. No.5,568,799, and will not be described in detail below.

A sensor output selector 24 shown in FIG. 1 serves to select the outputKACT from the LAF sensor 5, which is suitable for the estimation of areal air-fuel ratio #nA/F of each cylinder with the observer 21,depending on the operating state of the internal combustion engine 1.Details of the sensor output selector 24 are disclosed in detail inJapanese laid-open patent publication No. 7-259588 and U.S. Pat. No.5,540,209, and will not be described in detail below.

The exhaust-side control unit 7 a has a subtractor 11 for determining adifference kact (=KACT−FLAF/BASE) between the output KACT from the LAFsensor 5 and a predetermined reference value FLAF/BASE and a subtractor12 for determining a difference VO₂(=VO2/OUT−VO2/TARGET) between theoutput VO2/OUT from the O₂ sensor 6 and a target value VO2/TARGETtherefor.

The target value VO2/TARGET for the output VO2/OUT from the O₂ sensor 6is sequentially determined by the exhaust-side control unit 7 adepending on the operating state of the internal combustion engine 1 asan output value of the O₂ sensor 6 in order to achieve an optimumpurifying capability of the catalytic converter 3. In the presentembodiment, the exhaust-side control unit 7 a determines the targetvalue VO2/TARGET in each control cycle from the rotational speed NE andintake pressure PB of the internal combustion engine 1 using apredetermined map. The target value VO2/TARGET is a value between theoutput values VO2/OUT2, VO2/OUT3 shown in FIG. 2, for example.

In the present embodiment, the reference value FLAF/BASE with respect tothe output KACT from the LAF sensor 5 is set to a “stoichiometricair-fuel ratio”. The differences kact, VO₂ determined respectively bythe subtractors 11, 12 are referred to as a differential output kact ofthe LAF sensor 5 and a differential output VO2 of the O₂ sensor 6,respectively.

The exhaust-side control unit 7 a also has an exhaust-side mainprocessor 13 which is supplied with the data of the differential outputskact, VO2 as the data of the output from the LAF sensor 5 and the outputof the O₂ sensor 6.

As shown in FIG. 3, the exhaust-side main processor 13 comprises, as itsfunctions, a target air-fuel ratio calculating means 13 a forsequentially determining a target air-fuel ratio KCMD for the internalcombustion engine 1 based on the data of the differential outputs kact,VO2, and a deteriorated state evaluating means 13 b for evaluating thedeteriorated state of the catalytic converter 3 based on the data of thedifferential output VO2 of the O₂ sensor 6.

The target air-fuel ratio calculating means 13 a serves to control anobject exhaust system (denoted by E in FIG. 1) including the catalyticconverter 3, which ranges from the LAF sensor 5 to the O₂ sensor 6 alongthe exhaust pipe 2. The target air-fuel ratio calculating means 13 asequentially determines the target air-fuel ratio KCMD for the internalcombustion engine 1 so as to converge the output VO2/OUT of the O₂sensor 6 to the target value VO2/TARGET therefor according to a slidingmode control process, or specifically an adaptive sliding mode controlprocess, in view of a dead time present in the object exhaust system Eto be controlled, a dead time present in the internal combustion engine1 and the engine-side control unit 7 b, and behavioral changes of theobject exhaust system E.

The deteriorated state evaluating means 13 b serves to evaluate thedeteriorated state of the catalytic converter 3 based on the value of adeterioration evaluating linear function, described later on, which isdetermined from time-series data of the differential output VO2 of theO₂ sensor 6, and control the operation of a deterioration indicator 29(see FIG. 1) connected to the apparatus according to the presentembodiment depending on the evaluation of the deteriorated state of thecatalytic converter 3. The deterioration indicator 29 may comprise alamp, a buzzer, or a display unit for displaying characters, a graphicimage, etc. to indicate the deteriorated state of the catalyticconverter 3.

The target air-fuel ratio calculating means 13 a combined with theengine-side control unit 7 b corresponds to an air-fuel ratiomanipulating means according to the present invention. The deterioratedstate evaluating means 13 b includes the function of a parametergenerating means according to the present invention.

The target air-fuel ratio calculating means 13 a and the deterioratedstate evaluating means 13 b will further be described below.

In order to carry out the process of the target air-fuel ratiocalculating means 13 a, according to present embodiment, the objectexhaust system E is regarded as a system for generating the outputVO2/OUT of the O₂ sensor 6 (the detected value of the oxygenconcentration of the exhaust gas having passed through the catalyticconverter 3) from the output KACT of the LAF sensor 5 (the detectedvalue of the air-fuel ratio of the internal combustion engine 1) via adead time element and a response delay element, and the behavior of thesystem is modeled as a discrete time system.

In addition, the system comprising the internal combustion engine 1 andthe engine-side control unit 7 b is regarded as a system (hereinafterreferred to as “air-fuel ratio manipulating system”) for generating theoutput KACT of the LAF sensor 5 from the target air-fuel ratio KCMD viaa dead time element, and the behavior of this system is modeled as adiscrete time system.

In the present embodiment, the model (hereinafter referred to as“exhaust system model”) in which the behavior of the object exhaustsystem E is expressed as a discrete time system is expressed, using thedifferential output kact (=KACT−FLAF/BASE) from the LAF sensor 5 and thedifferential output VO2 (=VO2/OUT−VO2/TARGET) from the O₂ sensor 6,instead of the output KACT of the LAF sensor 5 and the output VO2/OUT ofthe O₂ sensor 6, according to the following equation (1):

VO 2(k+1)=a 1·VO 2(k)+a 2·VO 2(k−1)+b 1·kact(k−d 1)   (1)

According to the equation (1), the object exhaust system E is regardedas a system for generating the differential output VO2 from the O₂sensor 6 from the differential output kact from the LAF sensor 5 via adead time element and a response delay element, and expresses thebehavior of the object exhaust system E with the model of a discretetime system (more specifically, an autoregressive model having a deadtime in the differential output kact as an input to the exhaust system

In the equation (1), “k” represents the ordinal number of adiscrete-time control cycle of the exhaust-side control unit 7 a, and“d1” the dead time (more specifically, the dead time required until theair-fuel ratio detected at each point of time by the LAF sensor 5 isreflected in the output VO2/OUT of the O₂ sensor 6) of the objectexhaust system E as represented by the number of control cycles. Thedead time of the object exhaust system E is generally equal to the timeof 3-10 control cycles (d1=3-10) if the period (constant in the presentembodiment) of control cycles of the exhaust-side control unit 7 aranges from 30 to 100 ms. In the present embodiment, a preset constantvalue (d1=7, for example) which is equal to or slightly longer than theactual dead time of the object exhaust system E is used as the dead timed1 in the exhaust system model as represented by the equation (1).

The first and second terms of the right side of the equation (1)correspond to a response delay element of the object exhaust system E,the first term being a primary autoregressive term and the second termbeing a secondary autoregressive term. In the first and second terms,“a1”, “a2” represent respective gain coefficients of the primaryautoregressive term and the secondary autoregressive term. Statedotherwise, these gain coefficients a1, a2 are relative to thedifferential output VO2 of the O₂ sensor 6 as an output of the objectexhaust system E.

The third term of the right side of the equation (1) represents thedifferential output kact of the LAF sensor 5 as an input to the objectexhaust system E, including the dead time d1 of the object exhaustsystem E. In the third term, “b1” represents a gain coefficient relativeto the input to the object exhaust system E, i.e., the differentialoutput kact of the LAF sensor 5. These gain coefficients “a1”, “a2”,“b1” are parameters which define the behavior of the exhaust systemmodel, and are sequentially identified by an identifier which will bedescribed later on.

The model (hereinafter referred to as “air-fuel ratio manipulatingsystem model”) of the discrete time system of the air-fuel ratiomanipulating system which comprises the internal combustion engine 1 andthe engine-side control unit 7 b is expressed, using the differentialoutput kact (=KACT−FLAF/BASE) from the LAF sensor 5 instead of theoutput KACT from the LAF sensor 5 and also using a difference kcmd(=KCMD−FLAF/BASE, which corresponds to a target value for thedifferential output kact of the LAF sensor 5, and will be referred to as“target differential air-fuel ratio kcmd”) between the target air-fuelratio KCMD and the reference value FLAF/BASE instead of the targetair-fuel ratio KCMD, according to the following equation (2):

kact(k)=kcmd(k−d 2)   (2)

The equation (2) expresses the air-fuel ratio manipulating system as themodel of a discrete time system, regarding the air-fuel ratiomanipulating system as a system for generating the differential outputkact from the LAF sensor 5 from the target differential air-fuel ratiokcmd via a dead time element, i.e., a system in which the differentialoutput kact in each control cycle is equal to the target differentialair-fuel ratio kcmd prior to the dead time.

In the equation (2), “d2” represents the dead time of the air-fuel ratiomanipulating system (more specifically, the time required until thetarget air-fuel ratio KCMD at each point of time is reflected in theoutput signal KACT of the LAF sensor 5) in terms of the number ofcontrol cycles of the exhaust-side control unit 7 a. The dead time ofthe air-fuel ratio manipulating system varies with the rotational speedNE of the internal combustion engine 1, and is longer as the rotationalspeed NE of the internal combustion engine 1 is lower. In the presentembodiment, in view of the above characteristics of the dead time of theair-fuel ratio manipulating system, a preset constant value (forexample, d2=3) which is equal to or slightly longer than the actual deadtime of the air-fuel ratio manipulating system at an idling rotationalspeed of the internal combustion engine 1, which is a rotational speedin a low speed range of the internal combustion engine 1 (the actualdead time is a maximum dead time which can be taken by the air-fuelratio manipulating system at an arbitrary rotational speed of theinternal combustion engine 1), is used as the value of the dead time d2in the air-fuel ratio manipulating system model expressed by theequation (2).

The air-fuel ratio manipulating system actually includes a dead timeelement and a response delay element of the internal combustion engine1. Since a response delay of the output KACT of the LAF sensor 5 withrespect to the target air-fuel ratio KCMD is basically compensated forby the feedback controller 14 (particularly the adaptive controller 18)of the engine-side control unit 7 b, there will arise no problem if aresponse delay element of the internal combustion engine 1 is not takeninto account in the air-fuel ratio manipulating system as viewed fromthe exhaust-side control unit 7 a.

The target air-fuel ratio calculating means 13 a of the exhaust-sidemain processor 13 carries out the process for calculating the targetair-fuel ratio KCMD based on the exhaust system model and the air-fuelratio manipulating system model expressed respectively by the equations(1), (2) in control cycles of the exhaust-side control unit 7 a. Inorder to carry out the above process, the target air-fuel ratiocalculating means 13 a has its functions as shown in FIG. 3.

As shown in FIG. 3, the target air-fuel ratio calculating means 13 acomprises an identifier 25 for sequentially determining in each controlcycle identified values a1 hat, a2 hat, b1 hat of the gain coefficientsa1, a2, b1 (hereinafter referred to as “identified gain coefficients a1hat, a2 hat, b1 hat”) that are parameters to be established for theexhaust system model (the equation (1)), an estimator 26 forsequentially determining in each control cycle an estimated value VO2bar of the differential output VO2 from the O₂ sensor 6 (hereinafterreferred to as “estimated differential output VO2 bar”) after the totaldead time d (=d1+d2) which is the sum of the dead time d1 of the objectexhaust system E and the dead time d2 of the air-fuel ratio manipulatingsystem, and a sliding mode controller 27 for sequentially determining ineach control cycle a target air-fuel ratio KCMD according to an adaptiveslide mode control process.

The algorithm of a processing operation to be carried out by theidentifier 25, the estimator 26, and the sliding mode controller 27 isconstructed as follows:

The identifier 25 serves to identify the gain coefficients a1, a2, b1sequentially on a real-time basis for the purpose of minimizing amodeling error of the actual object exhaust system E of the exhaustsystem model expressed by the equation (1). The identifier 25 carriesout its identifying process as follows:

In each control cycle of the exhaust-side control unit 7 a, theidentifier 25 determines an identified value VO2(k) hat of thedifferential output VO2 (the output of the exhaust system model) fromthe O₂ sensor 6 (hereinafter referred to as “identified differentialoutput VO2(k) hat”) on the exhaust system model, using the identifiedgain coefficients a1 hat, a2 hat, b1 hat of the presently establishedexhaust system model, i.e., identified gain coefficients a1(k−1) hat,a2(k−1) hat, b1(k−1) hat determined in a preceding control cycle, andpast data kact(k−d1−1), VO2(k−1), VO2(k−2) of the differential outputkact from the LAF sensor 5 and the differential output VO2 from the O₂sensor 6, according to the following equation (3): $\begin{matrix}{{V\hat{O}2(k)} = \quad {{{\hat{a1}\left( {k - 1} \right)} \cdot {{VO2}\left( {k - 1} \right)}} + {{\hat{a2}\left( {k - 1} \right)} \cdot {{VO2}\left( {k - 2} \right)}} + {{\hat{b1}\left( {k - 1} \right)} \cdot {{kact}\left( {k - {d1} - 1} \right)}}}} & (3)\end{matrix}$

The equation (3) corresponds to the equation (1) which is shifted intothe past by one control cycle with the gain coefficients a1, a2, b1being replaced with the respective identified gain coefficients a1(k−1)hat, a2(k−1) hat, b1(k−1) hat. The constant value (d1=7) established asdescribed above is used as the value of the dead time d1 of the objectexhaust system E in the third term of the equation (3).

If vectors Θ, ξ defined by the following equations (4), (5) areintroduced (the letter T in the equations (4), (5) represents atransposition), then the equation (3) is expressed by the equation (6):$\begin{matrix}{{\Theta^{T}(k)} = \left\lceil {{\hat{a1}(k)}{\hat{a2}(k)}{\hat{b1}(k)}} \right\rceil} & (4)\end{matrix}$

ξ^(T)(k)=[VO 2(k−1)VO 2(k−2)kact(k−d1−1)]  (5)

VÔ 2(k)=Θ^(T)(k−1)·ξ(k)   (6)

The identifier 25 also determines a difference id/e(k) between theidentified differential output VO2(k) hat from the O₂ sensor 6 which isdetermined by the equation (3) or (6) and the present differentialoutput VO2(k) from the O₂ sensor 6, as representing a modeling error ofthe exhaust system model with respect to the actual object exhaustsystem E (hereinafter the difference id/e will be referred to as“identified error id/e”), according to the following equation (7):

id/e(k)=VO 2(k)−VÔ 2(k)   (7)

The identifier 25 further determines new identified gain coefficientsa1(k) hat, a2(k) hat, b1(k) hat, stated otherwise, a new vector Θ(k)having these identified gain coefficients as elements (hereinafter thenew vector Θ(k) will be referred to as “identified gain coefficientvector Θ”), in order to minimize the identified error id/e, according tothe equation (8) given below. That is, the identifier 25 varies theidentified gain coefficients a1 hat (k−1), a2 hat (k−1), b1 hat (k−1)determined in the preceding control cycle by a quantity proportional tothe identified error id/e for thereby determining the new identifiedgain coefficients a1(k) hat, a2(k) hat, b1(k) hat.

Θ(k)=Θ(k−1)+Kθ(k)·id/e(k)   (8)

where Kθ represents a cubic vector determined by the following equation(9), i.e., a gain coefficient vector for determining a change dependingon the identified error id/e of the identified gain coefficients a1 hat,a2 hat, b1 hat): $\begin{matrix}{{K\quad {\theta (k)}} = \frac{{P\left( {k - 1} \right)} \cdot {\xi (k)}}{1 + {{\xi^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi (k)}}}} & (9)\end{matrix}$

where P represents a cubic square matrix determined by a recursiveformula expressed by the following equation (10): $\begin{matrix}{{P(k)} = {\frac{1}{\lambda_{1}(k)} \cdot \left\lceil {I - \frac{{\lambda_{2}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi (k)} \cdot {\xi^{T}(k)}}{{\lambda_{1}(k)} + {{\lambda_{2}(k)} \cdot {\xi^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi (k)}}}} \right\rceil \cdot {P\left( {k - 1} \right)}}} & (10)\end{matrix}$

where I represents a unit matrix.

In the equation (10), λ₁, λ₂ are established to satisfy the conditions0<λ₁≦1 and 0≦λ₂<2, and an initial value P(0) of P represents a diagonalmatrix whose diagonal components are positive numbers.

Depending on how λ₁, λ₂ in the equation (10) are established, any one ofvarious specific algorithms including a fixed gain method, a degressivegain method, a method of weighted least squares, a method of leastsquares, a fixed tracing method, etc. may be employed. According to thepresent embodiment, a method of least squares (λ₁=λ₂=1), for example, isemployed.

Basically, the identifier 25 sequentially determines in each controlcycle the identified gain coefficients a1 hat, a2 hat, b1 hat of theexhaust system model in order to minimize the identified error id/eaccording to the above algorithm (calculating operation). Through thisoperation, it is possible to sequentially obtain the identified gaincoefficients a1 hat, a2 hat, b1 hat which match the actual objectexhaust system E.

The algorithm described above is the basic algorithm that is carried outby the identifier 25.

The estimator 26 sequentially determines in each control cycle theestimated differential output VO2 bar which is an estimated value of thedifferential output VO2 from the O₂ sensor 6 after the total dead time d(=d1+d2) in order to compensate for the effect of the dead time d1 ofthe object exhaust system E and the effect of the dead time d2 of theair-fuel ratio manipulating system for the calculation of the targetair-fuel ratio KCMD with the sliding mode controller 27 as described indetail later on. The algorithm for the estimator 26 to determine theestimated differential output VO2 bar is constructed as follows:

If the equation (2) expressing the model of the air-fuel ratiomanipulating system is applied to the equation (1) expressing the modelof the object exhaust system E, then the equation (1) can be rewrittenas the following equation (11): $\begin{matrix}\begin{matrix}{{{VO2}\left( {k + 1} \right)} = \quad {{{a1} \cdot {{VO2}(k)}} + {{a2} \cdot {{VO2}\left( {k - 1} \right)}} +}} \\{\quad {{b1} \cdot {{kcmd}\left( {k - {d1} - {d2}} \right)}}} \\{= \quad {{{a1} \cdot {{VO2}(k)}} + {{a2} \cdot {{VO2}\left( {k - 1} \right)}} + {{b1} \cdot {{kcmd}\left( {k - d} \right)}}}}\end{matrix} & (11)\end{matrix}$

The equation (11) expresses the behavior of a system which is acombination of the object exhaust system E and the air-fuel manipulatingsystem as the model of a discrete time system, regarding such a systemas a system for generating the differential output VO2 from the O₂sensor 6 from the target differential air-fuel ratio kcmd via dead timeelements of the object exhaust system E and the air-fuel manipulatingsystem and a response delay element of the object exhaust system E.

By using the equation (11), the estimated differential output VO2(k+d)bar which is an estimated value of the differential output VO2(k+d) ofthe O₂ sensor 6 after the total dead time d in each control cycle can beexpressed using time-series data VO2(k), VO2(k−1) of present and pastvalues of the differential output VO2 of the O₂ sensor 6 and time-seriesdata kcmd(k−j) (j=1, 2, . . . , d) of the past values of the targetdifferential air-fuel ratio kcmd (=KCMD−FLAF/BASE) which corresponds tothe target air-fuel ratio KCMD determined by the sliding mode controller27 (its specific process of determining the target air-fuel ratio KCMDwill be described later on), according to the following equation (12):$\begin{matrix}\begin{matrix}{{\overset{\_}{VO2L}\left( {k + d} \right)} = \quad {{\alpha \quad {1 \cdot {{VO2}(k)}}} + {{{\alpha 2} \cdot {VO2}}\left( {k - 1} \right)} +}} \\{\quad {\sum\limits_{j = 1}^{d}{\beta \quad {j \cdot {{kcmd}\left( {k - j} \right)}}}}\quad}\end{matrix} & (12)\end{matrix}$

where

α1=the first-row, first-column element of A^(d),

α2=the first-row, second-column element of A^(d),

βj=the first-row elements of A^(j−1)·B $A = \left\lceil \begin{matrix}{a1} & {a2} \\1 & 0\end{matrix} \right\rceil$ $B = \left\lceil \begin{matrix}{b1} \\0\end{matrix} \right\rceil$ $B = \begin{bmatrix}{b1} \\0\end{bmatrix}$

In the equation (12), “α1”, “α2” represent the first-row, first-columnelement and the first-row, second-column element, respectively, of thedth power A^(d) (d: total dead time) of the matrix A defined asdescribed above with respect to the equation (12), and “βj” (j=1, 2, . .. , d) represents the first-row elements of the product A^(j−1)·B of the(j−1)th power A^(j−1) (j=1, 2, . . . , d) of the matrix A and the vectorB defined as described above with respect to the equation (12).

Of the time-series data of the past values of the target combineddifferential air-fuel ratio kcmd according to the equation (12), thetime-series data kcmd(k−d2), kcmd(k−d2−1), . . . , kcmd(k−d) from thepresent prior to the dead time d2 of the air-fuel manipulating systemcan be replaced respectively with data kact(k), kact(k−1), . . . ,kact(k−d+d2) obtained prior to the present time of the differentialoutput kact of the LAF sensor 5 according the above equation (2). Whenthe time-series data are thus replaced, the following equation (13) isobtained: $\begin{matrix}\begin{matrix}{{\overset{\_}{VO2L}\left( {k + d} \right)} = \quad {{\alpha \quad {1 \cdot {{VO2}(k)}}} + {{{\alpha 2} \cdot {VO2}}\left( {k - 1} \right)} +}} \\{\quad {{\sum\limits_{j = 1}^{{d2} - 1}{\beta \quad {j \cdot {{kcmd}\left( {k - j} \right)}}}} + {\sum\limits_{i = 0}^{d - {d2}}{\beta \quad i}} + {{d2} \cdot {{kact}\left( {k - i} \right)}}}\quad} \\{= \quad {{\alpha \quad {1 \cdot {{VO2}(k)}}} + {{\alpha 2} \cdot {{VO2}\left( {k - 1} \right)}} +}} \\{\quad {{\sum\limits_{j = 1}^{{d2} - 1}{\beta \quad {j \cdot {{kcmd}\left( {k - j} \right)}}}} + {\sum\limits_{i = 0}^{d1}{\beta \quad i}} + {{d2} \cdot {{kact}\left( {k - i} \right)}}}\quad}\end{matrix} & (13)\end{matrix}$

The equation (13) is a basic formula for the estimator 26 to determinethe estimated differential output VO2(k+d) bar. Stated otherwise, theestimator 26 determines, in each control cycle, the estimateddifferential output VO2(k+d) bar of the O₂ sensor 6 according to theequation (13), using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6, the time-series datakcmd(k−j) (j=1, . . . , d2−1) of the past values of the targetdifferential air-fuel ratio kcmd which represents the target air-fuelratio KCMD determined in the past by the sliding mode controller 27, andthe time-series data kact(k−i) (i=0, . . . , d1) of the present and pastvalues of the differential output kact of the LAF sensor 5.

In the present embodiment, the values of the coefficients α1, α2, βj(j=1, 2, . . . , d) required to calculate the estimated differentialoutput VO2(k+d) bar according to the equation (13) are basicallycalculated using the identified gain coefficients a1 hat, a2 hat, b1 hatwhich are the identified values of the gain coefficients a1, a2, b1(which are elements of the vectors A, B defined with respect to theequation (12)). The values of the dead times d1, d2 required in theequation (13) comprise the preset values as described above.

The estimated differential output VO2(k+d) bar may be determinedaccording to the equation (12) without using the data of thedifferential output kact of the LAF sensor 5. For increasing thereliability of the estimated differential output VO2(k+d) bar, however,it is preferable to determine the estimated differential output VO2(k+d)bar according to the equation (13) using the data of the differentialoutput kact of the LAF sensor 5 which reflects the actual behavior ofthe internal combustion engine 1. If the dead time d2 of the air-fuelratio manipulating system can be set to “1”, all the time-series datakcmd(k−j) (j=1, 2, . . . , d) of the past values of the targetdifferential air-fuel ratio kcmd in the equation (12) may be replacedwith the time-series data kact(k), kact(k−1), . . . , kact(k−d+d2),respectively, prior to the present time, of the differential output kactof the LAF sensor 5. In this case, the estimated differential outputVO2(k+d) bar can be determined according to the following equation (14)which does not include the data of the target differential air-fuelratio kcmd: $\begin{matrix}\begin{matrix}{{\overset{\_}{VO2L}\left( {k + d} \right)} = \quad {{\alpha \quad {1 \cdot {{VO2}(k)}}} + {{{\alpha 2} \cdot {VO2}}\left( {k - 1} \right)} +}} \\{\quad {{\sum\limits_{j = 0}^{d - 1}{\beta \quad j}} + {1 \cdot {{kact}\left( {k - j} \right)}}}\quad}\end{matrix} & (14)\end{matrix}$

The sliding mode controller 27 will be described in detail below.

The sliding mode controller 27 determines an input quantity to be givento the object exhaust system E (which is specifically a target value forthe difference between the output KACT of the LAF sensor 5 (the detectedvalue of the air-fuel ratio) and the reference value FLAF/BASE, whichtarget value is equal to the target differential air-fuel ratio kcmd,the input quantity will be referred to as “SLD manipulating input Usl”)in order to cause the output VO2/OUT of the O₂ sensor 6 to settle on thetarget value VO2/TARGET, i.e., to converge the differential output VO2of the O₂ sensor 6 to “0” according to an adaptive sliding mode controlprocess which incorporates an adaptive control law for minimizing theeffect of a disturbance, in a normal sliding mode control process, anddetermines the target air-fuel ratio KCMD from the determined SLDmanipulating input Usl. An algorithm for carrying out the adaptivesliding mode control process is constructed as follows:

A switching function required for the algorithm of the adaptive slidingmode control process carried out by the sliding mode controller 27 and ahyperplane defined by the switching function (also referred to as a slipplane) will first be described below.

According to a basic concept of the sliding mode control process, thedifferential output VO2(k) of the O₂ sensor 6 obtained in each controlcycle and the differential output VO2(k−1) obtained in a precedingcontrol cycle are used as a state quantity to be controlled (controlledquantity), and a switching function σ for the sliding mode controlprocess is defined as a linear function whose variable components arerepresented by the differential outputs VO2(k), VO2(k−1) according tothe following equation (15): $\begin{matrix}{{{\sigma (k)} = {{{{s1} \cdot {{VO2}(k)}} + {{s2} \cdot {{VO2}\left( {k - 1} \right)}}} = {S \cdot X}}}{where}{{S = \left\lbrack {{s1}\quad {s2}} \right\rbrack},{X = \left\lceil \begin{matrix}{{VO2}(k)} \\{{VO2}\left( {k - 1} \right)}\end{matrix} \right\rceil}}} & (15)\end{matrix}$

A vector X defined above with respect to the equation (15) as a vectorwhose elements are represented by the differential outputs VO2(k),VO2(k−1) will hereinafter be referred to as a state quantity X.

The coefficients s1, s2 of the switching function σ are set in order tomeet the condition of the following equation (16): $\begin{matrix}{{- 1} < \frac{s2}{s1} < 1} & (16)\end{matrix}$

(when s1=1, −1<s2<1)

In the present embodiment, for the sake of brevity, the coefficient s1is set to s1=1 (s2/s1=s2), and the coefficient s2 is established tosatisfy the condition: −1<s2<1.

With the switching function σ thus defined, the hyperplane for thesliding mode control process is defined by the equation σ=0. Since thestate quantity X is of the second degree, the hyperplane σ=0 isrepresented by a straight line as shown in FIG. 4. At this time, thehyperplane is called a switching line or a switching plane depending onthe degree of a topological space.

In the present embodiment, the time-series data of the estimateddifferential output VO2 bar determined by the estimator 26 is used asthe variable components of the switching function for the sliding modecontrol process, as described later on.

The adaptive sliding mode control process serves to converge the statequantity X onto the hyperplane σ=0 according to a reaching control lawwhich is a control law for converging the state quantity X (=VO2(k),VO2(k−1)) onto the hyperplane σ=0, and an adaptive control law (adaptivealgorithm) which is a control law for compensating for the effect of adisturbance in converging the state quantity X onto the hyperplane σ=0(mode 1 in FIG. 4). While holding the state quantity X onto thehyperplane σ=0 according to an equivalent control input, the statequantity X is converged to a balanced point on the hyperplane σ=0 whereVO2(k)=VO2(k−1)=0, i.e., a point where time-series data VO2/OUT(k),VO2/OUT(k−1) of the output VO2/OUT of the O₂ sensor 6 are equal to thetarget value VO2/TARGET (mode 2 in FIG. 4).

The SLD manipulating input Usl (=the target differential air-fuel ratiokcmd) to be generated by the sliding mode controller 27 for convergingthe state quantity X toward the balanced point on the hyperplane σ=0 isexpressed as the sum of an equivalent control input Ueq to be applied tothe object exhaust system E according to the control law for convergingthe state quantity X onto the hyperplane σ=0, an input Urch (hereinafterreferred to as “reaching control law input Urch”) to be applied to theobject exhaust system E according to the reaching control law, and aninput Uadp (hereinafter referred to as “adaptive control law Uadp”) tobe applied to the object exhaust system E according to the adaptivecontrol law (see the following equation (17)).

Usl=Ueq+Urch+Uadp   (17)

The equivalent control input Ueq, the reaching control law input Urch,and the adaptive control law input Uadp are determined on the basis ofthe model of the discrete time system expressed by the equation (11),i.e., the model in which the differential output kact(k−d1) of the LAFsensor 5 in the equation (11) is replaced with the target differentialair-fuel ratio kcmd(k−d) using the total dead time d, as follows:

The equivalent control input Ueq which is an input component to beapplied to the object exhaust system E for converging the state quantityX onto the hyperplane σ=0 is the target differential air-fuel ratio kcmdwhich satisfies the condition: σ(k+1)=σ(k)=0. Using the equations (11),(15), the equivalent control input Ueq which satisfies the abovecondition is given by the following equation (18): $\begin{matrix}\begin{matrix}{{{Ueq}(k)} = \quad {{- \left( {S \cdot B} \right)^{- 1}} \cdot \left\{ {S \cdot \left( {A - 1} \right)} \right\} \cdot {X\left( {k + d} \right)}}} \\{= \quad {\frac{- 1}{{s1} \cdot {b1}} \cdot \left\{ {{\left\lbrack {{{s1} \cdot \left( {{a1} - 1} \right)} + {s2}} \right\rbrack \cdot {{VO2}\left( {k + d} \right)}} +} \right.}} \\{\quad \left. {\left( {{{s1} \cdot {a2}} - {s2}} \right) \cdot {{VO2}\left( {k + d - 1} \right)}} \right\}}\end{matrix} & (18)\end{matrix}$

The equation (18) is a basic formula for determining the equivalentcontrol law input Ueq(k) in each control cycle.

According to the present embodiment, the reaching control law input Urchis basically determined according to the following equation (19):$\begin{matrix}\begin{matrix}{{{Urch}(k)} = \quad {{- \left( {S \cdot B} \right)^{- 1}} \cdot F \cdot {\sigma \left( {k + d} \right)}}} \\{= \quad {\frac{- 1}{{s1} \cdot {b1}} \cdot F \cdot {\sigma \left( {k + d} \right)}}}\end{matrix} & (19)\end{matrix}$

Specifically, the reaching control law input Urch is determined inproportion to the value σ(k+d) of the switching function σ after thetotal dead time d, in view of the effect of the total dead time d.

The coefficient F in the equation (19) which determines the gain of thereaching control law is established to satisfy the condition expressedby the following equation (20):

0<F<2   (20)

The value of the switching function σ may possibly vary in anoscillating fashion (so-called chattering) with respect to thehyperplane σ=0. In order to suppress such chattering, it is preferablethat the coefficient F relative to the reaching control law input Urchbe established to further satisfy the condition of the followingequation (21):

0<F<1   (21)

The adaptive control law input Uadp is basically determined according tothe following equation (22) (ΔT in the equation (22) represents theperiod of the control cycles of the exhaust-side control unit 7 a):$\begin{matrix}\begin{matrix}{{{Uadp}(k)} = \quad {{- \left( {S \cdot B} \right)^{- 1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\sigma (i)} \cdot \Delta}\quad T} \right)}}} \\{= \quad {\frac{- 1}{{s1} \cdot {b1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\sigma (i)} \cdot \Delta}\quad T} \right)}}}\end{matrix} & (22)\end{matrix}$

The adaptive control law input Uadp is determined in proportion to anintegrated value (which corresponds to an integral of the values of theswitching function σ) over control cycles of the product of values ofthe switching function σ and the period ΔT of the exhaust-side controlunit 7 a until after the total dead time d, in view of the effect of thetotal dead time d.

The coefficient G (which determines the gain of the adaptive controllaw) in the equation (22) is established to satisfy the condition of thefollowing equation (23): $\begin{matrix}{{G = {J \cdot \frac{2 - F}{\Delta \quad T}}}\left( {0 < J < 2} \right)} & (23)\end{matrix}$

A specific process of deriving conditions for establishing the equations(16), (20), (21), (23) is described in detail in Japanese patentapplication No. 11-93741 and U.S. Pat. No. 6,082,099, and will not bedescribed in detail below.

In the present embodiment, the sliding mode controller 27 determines thesum (Ueq+Urch+Uadp) of the equivalent control input Ueq, the reachingcontrol law input Urch, and the adaptive control law Uadp determinedaccording to the respective equations (18), (19), (22) as the SLDmanipulating input Usl to be applied to the object exhaust system E.However, the differential outputs VO2(K+d), VO2(k+d−1) of the O₂ sensor6 and the value σ(k+d) of the switching function σ, etc. used in theequations (18), (19), (22) cannot directly be obtained as they arevalues in the future.

According to the present embodiment, therefore, the sliding modecontroller 27 actually uses the estimated differential outputs VO2(k+d)bar, VO2(k+d−1) bar determined by the estimator 26, instead of thedifferential outputs VO2(K+d), VO2(k+d−1) from the O₂ sensor 6 fordetermining the equivalent control input Ueq according to the equation(18), and calculates the equivalent control input Ueq in each controlcycle according to the following equation (24): $\begin{matrix}{{{Ueq}(k)} = {\frac{- 1}{{s1} \cdot {b1}}\left\{ {{\left\lbrack {{{s1} \cdot \left( {{a1} - 1} \right)} + {s2}} \right\rbrack \cdot {\overset{\_}{VO2}\left( {k + d} \right)}} + {\left( {{{s1} \cdot {a2}} - {s2}} \right) \cdot {\overset{\_}{VO2}\left( {k + d - 1} \right)}}} \right\}}} & (24)\end{matrix}$

According to the present embodiment, furthermore, the sliding modecontroller 27 actually uses time-series data of the estimateddifferential output VO2 bar sequentially determined by the estimator 26as described as a state quantity to be controlled, and defines aswitching function σ bar according to the following equation (25) (theswitching function σ bar corresponds to time-series data of thedifferential output VO2 in the equation (15) which is replaced withtime-series data of the estimated differential output VO2 bar), in placeof the switching function σ established according to the equation (15):

{overscore (σ(k))}=s 1·{overscore (VO 2)}(k)+s 2·{overscore (VO 2)}(k−1)  (25)

The sliding mode controller 27 calculates the reaching control law inputUrch in each control cycle according to the following equation (26),using the switching function σ bar represented by the equation (25),rather than the value of the switching function σ for determining thereaching control law input Urch according to the equation (19):$\begin{matrix}{{{Urch}(k)} = {\frac{- 1}{{s1} \cdot {b1}} \cdot F \cdot {\overset{\_}{\sigma}\left( {k + d} \right)}}} & (26)\end{matrix}$

Similarly, the sliding mode controller 27 calculates the adaptivecontrol law input Uadp in each control cycle according to the followingequation (27), using the value of the switching function σ barrepresented by the equation (25), rather than the value of the switchingfunction σ for determining the adaptive control law input Uadp accordingto the equation (22): $\begin{matrix}{{{{Uadp}(k)} = \frac{- 1}{{s1} \cdot {b1}}}{\cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\overset{\_}{\sigma}(i)} \cdot \Delta}\quad T} \right)}}} & (27)\end{matrix}$

The latest identified gain coefficients a1(k) hat, a2(k) hat, b1(k) hatwhich have been determined by the identifier 25 are basically used asthe gain coefficients a1, a1, b1 that are required to calculate theequivalent control input Ueq, the reaching control law input Urch, andthe adaptive control law input Uadp according to the equations (24),(26), (27).

The sliding mode controller 27 determines the sum of the equivalentcontrol input Ueq, the reaching control law input Urch, and the adaptivecontrol law input Uadp determined according to the equations (24), (26),(27), as the SLD manipulating input Usl to be applied to the objectexhaust system E (see the equation (17)). The conditions forestablishing the coefficients s1, s2, F, G used in the equations (24),(26), (27) are as described above.

The above process is a basic algorithm for determining the SLDmanipulating input Usl (=target differential air-fuel ratio kcmd) to beapplied to the object exhaust system E with the sliding mode controller27. According to the above algorithm, the SLD manipulating input Usl isdetermined to converge the estimated differential output VO2 bar fromthe O₂ sensor 6 toward “0”, and as a result, to convert the outputVO2/OUT from the O₂ sensor 6 toward the target value VO2/TARGET.

The sliding mode controller 27 eventually sequentially determines thetarget air-fuel ratio KCMD in each control cycle. The SLD manipulatinginput Usl determined as described above signifies a target value for thedifference between the air-fuel ratio of the exhaust gas detected by theLAF sensor 5 and the reference value FLAF/BASE, i.e., the targetdifferential air-fuel ratio kcmd. Consequently, the sliding modecontroller 27 eventually determines the target air-fuel ratio KCMD byadding the reference value FLAF/BASE to the determined SLD manipulatinginput Usl in each control cycle according to the following equation(28): $\begin{matrix}\begin{matrix}{{{KCMD}(k)} = \quad {{{Usl}(k)} + {{FLAF}/{BASE}}}} \\{= \quad {{{Ueq}(k)} + {{Urch}(k)} + {{Uadp}(k)} + {{FLAF}/{BASE}}}}\end{matrix} & (28)\end{matrix}$

The above process is a basic algorithm for determining the targetair-fuel ratio KCMD with the sliding mode controller 27 according to thepresent embodiment.

In the present embodiment, the stability of the adaptive sliding modecontrol process carried out by the sliding mode controller 27 is checkedfor limiting the value of the SLD manipulating input Usl. Details ofsuch a checking process will be described later on.

A process carried out by the deteriorated state evaluating means 13 bwill be described below.

When the target air-fuel ratio KCMD is sequentially determined by thesliding mode controller 27 and the fuel injection quantity of theinternal combustion engine 1 is adjusted by the engine-side control unit7 b is adjusted in order converge the output KACT (the detected value ofthe air-fuel ratio of the internal combustion engine 1) of the LAFsensor to the target air-fuel ratio KCMD, the time-series data of theoutput VO2/OUT of the O₂ sensor 6 exhibits characteristic changesdepending on the deteriorated state of the catalytic converter 3 withrespect to the hyperplane σ=0.

Such characteristic changes will be described below with reference toFIGS. 5 through 7. FIGS. 5 through 7 show sampled data, represented bystippled dots, of the set of the time-series data VO2(k), VO2(k−1) ofthe differential output VO2 of the O₂ sensor 6, i.e., the state quantityX, which are obtained in respective control cycles of the exhaust-sidecontrol unit 7 a when the fuel injection quantity of the internalcombustion engine 1 is adjusted depending on the target air-fuel ratioKCMD, with respect to a brand-new catalytic converter 3, a catalyticconverter 3 that has been deteriorated to a relatively small degree, anda catalytic converter 3 that has been deteriorated to a relatively largedegree. In each of FIGS. 5 through 7, the straight line represents thehyperplane σ=0.

As shown in FIG. 5, when the catalytic converter 3 is brand-new, thestate quantities X tend to concentrate in the vicinity of the hyperplaneσ=0. As the deterioration of the catalytic converter 3 progresses, asshown in FIGS. 6 and 7, the state quantities X tend to be distributedaway from the hyperplane σ=0, i.e., vary in a larger range around thehyperplane σ=0. The state quantities X tend to be distributed morewidely as the catalytic converter 3 is deteriorated to a larger degree,i.e., as the deterioration of the catalytic converter 3 progresses moregreatly. Stated otherwise, as the deterioration of the catalyticconverter 3 progresses, the switching function σ determined according tothe equation (15) is likely to have a value more remote from “0”, andhence vary more widely from “0”. This appears to be due to the fact thatas the deterioration of the catalytic converter 3 progresses, theexhaust system model according to the equation (1) tends to suffer anerror, and hence the ability of the state quantity X to converge to thehyperplane σ=0 is lowered.

The above tendency also is also exhibited by the value of the switchingfunction σ bar determined by the equation (25) where the estimateddifferential output VO2 bar determined by the estimator 26 is used as avariable component, i.e., the switching function used actually as theswitching function for the sliding mode control process, if the targetvalue VO2/TARGET for the output VO2/OUT of the O₂ sensor 6 is constant.Whereas the switching function σ bar employs the estimated value of thedifferential output VO2 of the O₂ sensor 6, the switching functionaccording to the equation (15) employs the actual differential outputVO2 of the O₂ sensor 6. Therefore, the latter appears to better reflectthe actual deteriorated state of the catalytic converter 3.

For the above reason, according to the present embodiment, adeterioration evaluating parameter for evaluating the deteriorated stateof the catalytic converter 3 is generated, as described later on, usingthe value of the switching function σ according to the equation (15),and the deteriorated state of the catalytic converter 3 is evaluatedbased on the value of the deterioration evaluating parameter. Asdescribed above, the actual switching function for the sliding modecontrol process is the switching function σ bar defined according to theequation (25) where the estimated differential output VO2 bar determinedby the estimator 26 is used as a variable component. Strictly, theswitching function σ according to the equation (15) is not the switchingfunction for the sliding mode control process according to the presentembodiment. The switching function σ according to the equation (15) willhereinafter be referred to as “deterioration evaluating linear functionσ”.

An algorithm for evaluating the deteriorated state of the catalyticconverter 3 based on the deterioration evaluating linear function σ withthe deteriorated state evaluating means 13 b is constructed as follows:

In view of the above tendency of the deterioration evaluating linearfunction σ to change as the deterioration of the catalytic converter 3progresses, the deteriorated state evaluating means 13 b sequentiallydetermines the square σ² of the value of the deterioration evaluatinglinear function σ in each control cycle.

Then, the deteriorated state evaluating means 13 b effects a low-passfiltering process on the square σ² to determine a central value of thesquare σ² (hereinafter represented by LSσ²) as a basic deteriorationevaluating parameter.

The above filtering process to determine the deterioration evaluatingparameter LSσ² comprises a sequential statistic processing algorithm,and is represented by the following equation (29): $\begin{matrix}{{{LS}\quad {\sigma^{2}(k)}} = {{{LS}\quad {\sigma^{2}\left( {k - 1} \right)}} + {\frac{{BP}\left( {k - 1} \right)}{1 + {{BP}\left( {k - 1} \right)}} \cdot \left( {{\sigma^{2}(k)} - {{LS}\quad {\sigma^{2}\left( {k - 1} \right)}}} \right)}}} & (29)\end{matrix}$

Thus, the deterioration evaluating parameter LSσ² is determined whilebeing sequentially updated, in each control cycle of the exhaust-sidecontrol unit 7 a, from a previous value LSσ²(k−1) of the deteriorationevaluating parameter LSσ², a present value σ²k of the square σ², and again parameter BP updated in each control cycle by a recursive formulaexpressed by the following equation (30): $\begin{matrix}{{{BP}(k)} = {\frac{1}{\eta 1} \cdot \left( {1 - \frac{{\eta 2} \cdot {{BP}\left( {k - 1} \right)}}{{\eta 1} + {{\eta 2} \cdot {{BP}\left( {k - 1} \right)}}}} \right) \cdot {{BP}\left( {k - 1} \right)}}} & (30)\end{matrix}$

In the equation (30), η1, η2 are set to values that satisfy theconditions: 0<η1≦1 and 0≦η2<1. Depending on how the values of η1, η2 areset, various specific algorithms including a fixed gain method, adegression method, a method of weighted least squares, a method of leastsquares, a fixed tracing method, etc. are constructed. According to thepresent embodiment, η1 is set to a given positive value smaller than “1”(0<η1<1), and η2=1, and the algorithm of the method of weighted leastsquares is employed.

When the deterioration evaluating parameter LSσ² as the central value(the central value of the minimum square in the present embodiment) ofthe square σ² of the deterioration evaluating linear function σ isdetermined, the value of the deterioration evaluating parameter LSσ²exhibits a tendency shown in FIG. 8 with respect to the deterioratedstate of the catalytic converter 3 (it is assumed that the target valueVO2/TARGET for the output VO2/OUT of the O₂ sensor 6 is constant). FIG.8 shows the relationship between the deterioration evaluating parameterLSσ² determined as described above and the rate of flow of the exhaustgas through the catalytic converter 3 (hereinafter referred to as“exhaust gas volume”) at the respective deteriorated states of thecatalytic converters 3 shown in FIGS. 5 through 7.

As shown in FIG. 8, the deterioration evaluating parameter LSσ² remainssubstantially constant irrespective of the exhaust gas volume at each ofthe deteriorated states of the catalytic converters 3, and increases itsvalue as the deterioration of the catalytic converters 3 progresses.Therefore, the deterioration evaluating parameter LSσ² represents thedegree to which the catalytic converter 3 is deteriorated.

In the present embodiment, the deteriorated state of the catalyticconverter 3 is evaluated to judge whether the catalytic converter 3 isin a state where it has been deteriorated to the extent that it needs tobe replaced immediately or soon (such a deteriorated state willhereinafter be referred to as “deterioration-in-progress state”, or not(a state of the catalytic converter 3 which is not in thedeterioration-in-progress state will hereinafter be referred to as“non-deteriorated state”). The deterioration-in-progress state isindicated by the deterioration indicator 29.

As indicated by the broken line in FIG. 8, a threshold CATAGELMT ispreset with respect to the deterioration evaluating parameter LSσ². Ifthe deterioration evaluating parameter LSσ² is equal to or greater thanthe threshold CATAGELMT, then the catalytic converter 3 is judged asbeing in the deterioration-in-progress state. If the deteriorationevaluating parameter LSσ² is smaller than the threshold CATAGELMT, thenthe catalytic converter 3 is judged as being in the non-deterioratedstate.

In the above description of the algorithm of the deteriorated stateevaluating means 13 b, it has been described that the target valueVO2/TARGET for the output VO2/OUT of the O₂ sensor 6 is constant. In thepresent embodiment, the target value VO2/TARGET is sequentially variablydetermined depending on the operating state, specifically, therotational speed NE and the intake pressure PB, of the internalcombustion engine 1. Since the output of the O₂ sensor 6 is non linear(see FIG. 2), the variation of the value of the deterioration evaluatinglinear function σ with respect to “0” while the control process is beingcarried out to converge the output of the O₂ sensor 6 to the targetvalue VO2/TARGET is affected by not only the deteriorated state of thecatalytic converter 3, but also the target value VO2/TARGET. Forexample, the sensitivity of the output of the O₂ sensor 6 is higher whenthe target value VO2/TARGET is VO2/TARGET=VO2/OUT1 than when the targetvalue VO2/TARGET is VO2/TARGET=VO2/OUT2, and hence the output of the O₂sensor 6 is likely to vary more with respect to the target valueVO2/TARGET when the target value VO2/TARGET is VO2/TARGET=VO2/OUT1. As aresult, even if the deteriorated state of the catalytic converter 3remains the same, the value of the deterioration evaluating linearfunction σ tends to vary to a greater extent when the target valueVO2/TARGET is VO2/TARGET=VO2/OUT1 than when the target value VO2/TARGETis VO2/TARGET=VO2/OUT2, resulting in a greater value of thedeterioration evaluating parameter LSσ².

According to the present embodiment, the deteriorated state evaluatingmeans 13 b calculates an average value VO2/OUTAVE of the output VO2/OUTof the O₂ sensor 6 while sequentially updating same, concurrent withcalculating the deterioration evaluating parameter LSσ₂ according to theequation (29). For determining the deteriorated state of the catalyticconverter 3, the deteriorated state evaluating means 13 b actuallycorrects the deterioration evaluating parameter LSσ² finally determinedaccording to the equation (29) depending on the average value VO2/OUTAVE(latest value) of the output VO2/OUT of the O₂ sensor 6, and comparesthe corrected deterioration evaluating parameter with the thresholdCATAGELMT to determine whether the catalytic converter 3 is in thedeterioration-in-progress state or the non-deteriorated state. In thedescription that follows, the deterioration evaluating parameter LSσ²before it is corrected will be referred to as “basic deteriorationevaluating parameter LSσ²”, and the deterioration evaluating parameterafter it is corrected will be referred to as “corrected deteriorationevaluating parameter CLSσ²”.

For example, the deterioration evaluating parameter LSσ² is corrected asfollows: The deteriorated state evaluating means 13 b determines acorrective coefficient KRESS (>0) for correcting the basic deteriorationevaluating parameter LSσ² based on a predetermined data table shown inFIG. 9, and multiplies the basic deterioration evaluating parameter LSσ²by the determined corrective coefficient KRESS, thereby determining acorrected deterioration evaluating parameter CLSσ². The data table shownin FIG. 9 is established such that the corrective coefficient KRESS isof a minimum value when the average value VO2/OUTAVE of the outputVO2/OUT of the O₂ sensor 6 is a value maximizing the sensitivity of theoutput VO2/OUT of the O₂ sensor 6, i.e., the level VO2/OUT1 shown inFIG. 2 (which is equivalent to the average value of the target valueVO2/TARGET being in the vicinity of the level VO2/OUT1), and isprogressively greater as the difference |VO2/OUTAVE−VO2OUT1| between theaverage value VO2/OUTAVE and the level VO2/OUT1 is greater.

The corrected deterioration evaluating parameter CLSσ² which is producedby multiplying the basic deterioration evaluating parameter LSσ² by thecorrective coefficient KRESS basically varies depending on only thedeteriorated state of the catalytic converter 3, rather than the targetvalue VO2/TARGET for the output VO2/OUT of the O² sensor 6.

The algorithm described above is a basic algorithm for evaluating thedeteriorated state of the catalytic converter 3 with the deterioratedstate evaluating means 13 b. The deteriorated state evaluating means 13b also performs an additional process of recognizing how the exhaust gasvolume changes upon evaluating the deteriorated state of the catalyticconverter 3. Such an additional process of recognizing how the exhaustgas volume changes will be described later on.

The general feedback controller 15 of the engine-side control unit 7 b,particularly, the adaptive controller 18, will further be describedbelow.

In FIG. 1, the general feedback controller 15 effects a feedback controlprocess to converge the output KACT (the detected value of the air-fuelratio) from the LAF sensor 5 toward the target air-fuel ratio KCMD asdescribed above. If such a feedback control process were carried outunder the known PID control only, it would be difficult to keep stablecontrollability against dynamic behavioral changes including changes inthe operating state of the internal combustion engine 1, characteristicchanges due to aging of the internal combustion engine 1, etc.

The adaptive controller 18 is a recursive-type controller which makes itpossible to carry out a feedback control process while compensating fordynamic behavioral changes of the internal combustion engine 1. As shownin FIG. 10, the adaptive controller 18 comprises a parameter adjuster 30for establishing a plurality of adaptive parameters using the parameteradjusting law proposed by I. D. Landau, et al., and a manipulatedvariable calculator 31 for calculating the feedback manipulated variableKSTR using the established adaptive parameters.

The parameter adjuster 30 will be described below. According to theparameter adjusting law proposed by I. D. Landau, et al., whenpolynomials of the denominator and the numerator of a transfer functionB(Z⁻¹)/A(Z⁻¹) of a discrete-system object to be controlled are generallyexpressed respectively by equations (31), (32), given below, an adaptiveparameter θ hat (j) (j indicates the ordinal number of a control cycle)established by the parameter adjuster 30 is represented by a vector(transposed vector) according to the equation (33) given below. An inputζ(j) to the parameter adjuster 30 is expressed by the equation (34)given below. In the present embodiment, it is assumed that the internalcombustion engine 1, which is an object to be controlled by the generalfeedback controller 15, is considered to be a plant of a first-ordersystem having a dead time dp corresponding to the time of threecombustion cycles of the internal combustion engine 1, and m=n=1, dp=3in the equations (31)-(34), and five adaptive parameters s0, r1, r2, r3,b0 are established (see FIG. 10). In the upper and middle expressions ofthe equation (34), us, ys generally represent an input (manipulatedvariable) to the object to be controlled and an output (controlledvariable) from the object to be controlled. In the present embodiment,the input is the feedback manipulated variable KSTR and the output fromthe object (the internal combustion engine 1) is the output KACT(detected air-fuel ratio) from the LAF sensor 4, and the input ζ(j) tothe parameter adjuster 30 is expressed by the lower expression of theequation (34) (see FIG. 10).

A(Z ⁻¹)=1+a 1 Z ⁻¹ + . . . + anZ ^(−n)   (31)

B(Z ⁻¹)=b 0+b 1 Z ⁻¹ + . . . +bmZ ^(−m)   (32) $\begin{matrix}\begin{matrix}{{{\hat{\theta}}^{T}(j)} = \quad \left\lbrack {{\hat{b}0(j)},{\hat{B}{R\left( {Z^{- 1},j} \right)}{\hat{S}\left( {Z^{- 1},j} \right)}}} \right\rbrack} \\{= \quad \left\lbrack {{{b0}(j)},{{r1}(j)},\cdots \quad,{{rm} + {dp} - {1(j)}},{{s0}(j)},\cdots \quad,{{sn} - {1(j)}}} \right\rbrack} \\{= \quad \left\lbrack {{{b0}(j)},{{r1}(j)},{{r2}(j)},{{r3}(j)},{{s0}(j)}} \right\rbrack}\end{matrix} & (33) \\\begin{matrix}{{\zeta^{T}(j)} = \quad \left\lbrack {{{us}(j)},\cdots \quad,{{us}\left( {j - m - {dp} + 1} \right)},{{ys}(j)},\cdots \quad,} \right.} \\\left. \quad {{ys}\left( {j - n + 1} \right)} \right\rbrack \\{= \quad \left\lbrack {{{us}(j)},{{us}\left( {j - 1} \right)},{{us}\left( {j - 2} \right)},{{us}\left( {j - 3} \right)},{{ys}(j)}} \right\rbrack} \\{= \quad \left\lbrack {{{KSTR}(j)},{{KSTR}\left( {j - 1} \right)},{{KSTR}\left( {j - 2} \right)},} \right.} \\\left. \quad {{{KSTR}\left( {j - 3} \right)},{{KACT}(j)}} \right\rbrack\end{matrix} & (34)\end{matrix}$

The adaptive parameter θ hat expressed by the equation (33) is made upof a scalar quantity element b0 hat (j) for determining the gain of theadaptive controller 18, a control element BR hat (Z⁻¹,j) expressed usinga manipulated variable, and a control element S (Z⁻¹,j) expressed usinga controlled variable, which are expressed respectively by the followingequations (35)˜(37) (see the block of the manipulated variablecalculator 31 shown in FIG. 10): $\begin{matrix}{{\hat{b}0^{- 1}(j)} = \frac{1}{b0}} & (35) \\\begin{matrix}{{\hat{B}{R\left( {Z^{- 1},j} \right)}} = {{r1Z}^{- 1} + {r2Z}^{- 2} + \ldots + {rm} + {dp} - {1Z^{- {({n + {dp} - 1})}}}}} \\{= {{r1Z}^{- 1} + {r2Z}^{- 2} + {r3Z}^{- 3}}}\end{matrix} & (36) \\\begin{matrix}{{\hat{S}\left( {Z^{- 1},j} \right)} = {{s0} + {s1Z}^{- 1} + \ldots + {sn} - {1Z^{- {({n - 1})}}}}} \\{= {s0}}\end{matrix} & (37)\end{matrix}$

The parameter adjuster 30 establishes coefficients of the scalarquantity element and the control elements, described above, and suppliesthem as the adaptive parameter θ hat expressed by the equation (33) tothe manipulated variable calculator 31. The parameter adjuster 30calculates the adaptive parameter θ hat so that the output KACT from theLAF sensor 5 will agree with the target air-fuel ratio KCMD, usingtime-series data of the feedback manipulated variable KSTR from thepresent to the past and the output KACT from the LAF sensor 5.

Specifically, the parameter adjuster 30 calculates the adaptiveparameter θ hat according to the following equation (38):

{circumflex over (θ)}(j)={circumflex over (θ)}(j−1)+Γ(j−1)·ζ(j−dp)·e*(j)  (38)

where Γ(j) represents a gain matrix (whose degree is indicated bym+n+dp) for determining a rate of establishing the adaptive parameter θhat, and e*(j) an estimated error of the adaptive parameter θ hat. Γ(j)and e*(j) are expressed respectively by the following recursive formulas(39), (40): $\begin{matrix}{{\Gamma (j)} = {\frac{1}{\lambda \quad 1(j)} \cdot \left\lceil {{\Gamma \left( {j - 1} \right)} - \frac{\lambda \quad 2{(j) \cdot {\Gamma \left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)} \cdot {\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma \left( {j - 1} \right)}}}{{{\lambda 1}(j)} + {{{\lambda 2}(j)} \cdot {\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma \left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)}}}} \right.}} & (39)\end{matrix}$

where 0<λ1(j)≦1, 0≦λ2(j)<2, Γ(0)>0. $\begin{matrix}{{e*(j)} = \frac{{{D\left( Z^{- 1} \right)} \cdot {{KACT}(j)}} - {{{\hat{\theta}}^{T}\left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)}}}{1 + {{\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma \left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)}}}} & (40)\end{matrix}$

where D(Z⁻¹) represents an asymptotically stable polynomial foradjusting the convergence. In the present embodiment, D(Z⁻¹)=1.

Various specific algorithms including the degressive gain algorithm, thevariable gain algorithm, the fixed tracing algorithm, and the fixed gainalgorithm are obtained depending on how λ1(j), λ2(j) in the equation(39) are selected. For a time-dependent plant such as a fuel injectionprocess, an air-fuel ratio, or the like of the internal combustionengine 1, either one of the degressive gain algorithm, the variable gainalgorithm, the fixed gain algorithm, and the fixed tracing algorithm issuitable.

Using the adaptive parameter θ hat (s0, r1, r2, r3, b0) established bythe parameter adjuster 30 and the target air-fuel ratio KCMD determinedby the deteriorated state evaluating means 13 b of the exhaust-side mainprocessor 13, the manipulated variable calculator 31 determines thefeedback manipulated variable KSTR according to a recursive formulaexpressed by the following equation (41): $\begin{matrix}{{{KSTR}(j)} = {\frac{1}{b0} \cdot \left\lbrack {{{KCMD}(j)} - {{s0} \cdot {{KACT}(j)}} - {{r1} \cdot {{KSTR}\left( {j - 1} \right)}} - {{r2} \cdot {{KSTR}\left( {j - 2} \right)}} - {{r3} \cdot {{KSTR}\left( {j - 3} \right)}}} \right\rbrack}} & (41)\end{matrix}$

The manipulated variable calculator 31 shown in FIG. 10 represents ablock diagram of the calculations according to the equation (41).

The feedback manipulated variable KSTR determined according to theequation (41) becomes the target air-fuel ratio KCMD insofar as theoutput KACT of the LAF sensor 4 agrees with the target air-fuel ratioKCMD. Therefore, the feedback manipulated variable KSTR is divided bythe target air-fuel ratio KCMD by the divider 19 for thereby determiningthe feedback manipulated variable kstr that can be used as the feedbackcorrection coefficient KFB.

As is apparent from the foregoing description, the adaptive controller18 thus constructed is a recursive-type controller taking into accountdynamic behavioral changes of the engine 1 which is an object to becontrolled. Stated otherwise, the adaptive controller 18 is a controllerdescribed in a recursive form to compensate for dynamic behavioralchanges of the engine 1, and more particularly a controller having arecursive-type adaptive parameter adjusting mechanism.

A recursive-type controller of this type may be constructed using anoptimum regulator. In such a case, however, it generally has noparameter adjusting mechanism. The adaptive controller 18 constructed asdescribed above is suitable for compensating for dynamic behavioralchanges of the internal combustion engine 1.

The details of the adaptive controller 18 have been described above.

The PID controller 17, which is provided together with the adaptivecontroller 18 in the general feedback controller 15, calculates aproportional term (P term), an integral term (I term), and a derivativeterm (D term) from the difference between the output KACT of the LAFsensor 5 and the target air-fuel ratio KCMD, and calculates the total ofthose terms as the feedback manipulated variable KLAF, as is the casewith the general PID control process. In the present embodiment, thefeedback manipulated variable KLAF is set to “1” when the output KACT ofthe LAF sensor 5 agrees with the target air-fuel ratio KCMD by settingan initial value of the integral term (I term) to “1”, so that thefeedback manipulated variable KLAF can be used as the feedbackcorrection coefficient KFB for directly correcting the fuel injectionquantity. The gains of the proportional term, the integral term, and thederivative term are determined from the rotational speed and intakepressure of the internal combustion engine 1 using a predetermined map.

The switcher 20 of the general feedback controller 15 outputs thefeedback manipulated variable KLAF determined by the PID controller 17as the feedback correction coefficient KFB for correcting the fuelinjection quantity if the combustion in the internal combustion engine 1tends to be unstable as when the temperature of the coolant of theinternal combustion engine 1 is low, the internal combustion engine 1rotates at high speeds, or the intake pressure is low, or if the outputKACT of the LAF sensor 5 is not reliable due to a response delay of theLAF sensor 5 as when the target air-fuel ratio KCMD changes largely orimmediately after the air-fuel ratio feedback control process hasstarted, or if the internal combustion engine 1 operates highly stablyas when it is idling and hence no high-gain control process by theadaptive controller 18 is required. Otherwise, the switcher 20 outputsthe feedback manipulated variable kstr which is produced by dividing thefeedback manipulated variable KSTR determined by the adaptive controller18 by the target air-fuel ration KCMD, as the feedback correctioncoefficient KFB for correcting the fuel injection quantity. This isbecause the adaptive controller 18 effects a high-gain control processand functions to converge the output KACT of the LAF sensor 5 quicklytoward the target air-fuel ratio KCMD, and if the feedback manipulatedvariable KSTR determined by the adaptive controller 18 is used when thecombustion in the internal combustion engine 1 is unstable or the outputKACT of the LAF sensor 5 is not reliable, then the air-fuel ratiocontrol process tends to be unstable.

Such operation of the switcher 20 is disclosed in detail in Japaneselaid-open patent publication No. 8-105345 or U.S. Pat. No. 5,558,075,and will not be described in detail below.

Operation of the apparatus according to the present embodiment will bedescribed below.

First, a process, carried out by the engine-side control unit 7 b, ofcalculating an output fuel injection quantity #nTout (n=1, 2, 3, 4) foreach of the cylinders of the internal combustion engine 1 forcontrolling the air-fuel ratio of the internal combustion engine 1 willbe described below with reference to FIG. 11. The engine-side controlunit 7 b calculates an output fuel injection quantity #nTout (n=1, 2, 3,4) for each of the cylinders in synchronism with a crankshaft angleperiod (TDC) of the internal combustion engine 1 as follows:

In FIG. 11, the engine-side control unit 7 b reads outputs from varioussensors including the LAF sensor 5 and the O₂ sensor 6 in STEPa. At thistime, the output KACT of the LAF sensor 5 and the output VO2/OUT of theO₂ sensor 6, including data obtained in the past, are stored in atime-series fashion in a memory (not shown).

Then, the basic fuel injection quantity calculator 8 corrects a fuelinjection quantity corresponding to the rotational speed NE and intakepressure PB of the internal combustion engine 1 depending on theeffective opening area of the throttle valve, thereby calculating abasic fuel injection quantity Tim in STEPb. The first correctioncoefficient calculator 9 calculates a first correction coefficientKTOTAL depending on the coolant temperature and the amount by which thecanister is purged in STEPc.

The engine-side control unit 7 b decides whether the operation mode ofthe internal combustion engine 1 is an operation mode (hereinafterreferred to as “normal operation mode”) in which the fuel injectionquantity is adjusted using the target air-fuel ratio KCMD generated bythe target air-fuel ratio calculating means 13 b, and sets a value of aflag f/prism/on in STEPd. When the value of the flag f/prism/on is “1”,it means that the operation mode of the internal combustion engine 1 isthe normal operation mode, and when the value of the flag f/prism/on is“0”, it means that the operation mode of the internal combustion engine1 is not the normal operation mode.

The deciding subroutine of STEPd is shown in detail in FIG. 12. As shownin FIG. 12, the engine-side control unit 7 b decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEPd-1, STEPd-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the exhaust-side main processor 13 are not accurateenough, the operation mode of the internal combustion engine 1 is notthe normal operation mode, and the value of the flag f/prism/on is setto “0” in STEPd-10.

Then, the engine-side control unit 7 b decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEPd-3. The engine-side control unit 7 b decides whether the ignitiontiming of the internal combustion engine 1 is retarded for earlyactivation of the catalytic converter 3 immediately after the start ofthe internal combustion engine 1 or not in STEPd-4. The engine-sidecontrol unit 7 b decides whether the throttle valve of the internalcombustion engine 1 is substantially fully open or not in STEPd-5. Theengine-side control unit 7 b decides whether the supply of fuel to theinternal combustion engine 1 is being stopped or not in STEPd-6. Ifeither one of the conditions of these steps is satisfied, then since itis not preferable or not possible to control the supply of fuel to theinternal combustion engine 1 using the target air-fuel ratio KCMDgenerated by the exhaust-side main processor 13, the operation mode ofthe internal combustion engine 1 is not the normal operation mode, andthe value of the flag f/prism/on is set to “0” in STEPd-10.

The engine-side control unit 7 b then decides whether the rotationalspeed NE and the intake pressure PB of the internal combustion engine 1fall within respective given ranges or not respectively in STEPd-7,STEPd-8. If either one of the rotational speed NE and the intakepressure PB does not fall within its given range, then since it is notpreferable to control the supply of fuel to the internal combustionengine 1 using the target air-fuel ratio KCMD generated by theexhaust-side main processor 13, the operation mode of the internalcombustion engine 1 is not the normal operation mode, and the value ofthe flag f/prism/on is set to “0” in STEPd-10.

If the conditions of STEPd-1, STEPd-2, STEPd-7, STEPd-8 are satisfied,and the conditions of STEPd-3, STEPd-4, STEPd-5, STEPd-6 are notsatisfied (at this time, the internal combustion engine 1 is in thenormal operation mode), then the operation mode of the internalcombustion engine 1 is judged as the normal operation mode, and thevalue of the flag f/prism/on is set to “1” in STEPd-9.

In FIG. 11, after the value of the flag f/prism/on has been set, theengine-side control unit 7 b determines the value of the flag f/prism/onin STEPe. If f/prism/on=1, then the engine-side control unit 7 b readsthe target air-fuel ratio KCMD generated by the exhaust-side mainprocessor 13 in STEPf. If f/prism/on=0, then the engine-side controlunit 7 b sets the target air-fuel ratio KCMD to a predetermined value inSTEPg. The predetermined value to be established as the target air-fuelratio KCMD is determined from the rotational speed NE and intakepressure PB of the internal combustion engine 1 using a predeterminedmap, for example.

In the local feedback controller 16, the PID controllers 22 calculatesrespective feedback correction coefficients #nKLAF in order to eliminatevariations between the cylinders, based on actual air-fuel ratios #nA/F(n=1, 2, 3, 4) of the respective cylinders which have been estimatedfrom the output KACT of the LAF sensor 5 by the observer 21, in STEPh.Then, the general feedback controller 15 calculates a feedbackcorrection coefficient KFB in STEPi.

Depending on the operating state of the internal combustion engine 1,the switcher 20 selects either the feedback manipulated variable KLAFdetermined by the PID controller 17 or the feedback manipulated variablekstr which has been produced by dividing the feedback manipulatedvariable KSTR determined by the adaptive controller 18 by the targetair-fuel ratio KCMD (normally, the switcher 20 selects the feedbackmanipulated variable kstr). The switcher 20 then outputs the selectedfeedback manipulated variable KLAF or kstr as a feedback correctioncoefficient KFB.

When switching the feedback correction coefficient KFB from the feedbackmanipulated variable KLAF from the PID controller 17 to the feedbackmanipulated variable kstr from the adaptive controller 18, the adaptivecontroller 18 determines a feedback manipulated variable KSTR in amanner to hold the correction coefficient KFB to the precedingcorrection coefficient KFB (=KLAF) as long as in the cycle time for theswitching. When switching the feedback correction coefficient KFB fromthe feedback manipulated variable kstr from the adaptive controller 18to the feedback manipulated variable KLAF from the PID controller 17,the PID controller 17 calculates a present correction coefficient KLAFin a manner to regard the feedback manipulated variable KLAF determinedby itself in the preceding cycle time as the preceding correctioncoefficient KFB (=kstr).

After the feedback correction coefficient KFB has been calculated, thesecond correction coefficient calculator 10 calculates in STEPj a secondcorrection coefficient KCMDM depending on the target air-fuel ratio KCMDdetermined in STEPf or STEPg.

Then, the engine-side control unit 7 b multiplies the basic fuelinjection quantity Tim determined as described above, by the firstcorrection coefficient KTOTAL, the second correction coefficient KCMDM,the feedback correction coefficient KFB, and the feedback correctioncoefficients #nKLAF of the respective cylinders, determining output fuelinjection quantities #nTout of the respective cylinders in STEPk. Theoutput fuel injection quantities #nTout are then corrected foraccumulated fuel particles on intake pipe walls of the internalcombustion engine 1 by the fuel accumulation corrector 23 in STEPm. Thecorrected output fuel injection quantities #nTout are applied to thenon-illustrated fuel injectors of the internal combustion engine 1 inSTEPn.

In the internal combustion engine 1, the fuel injectors inject fuel intothe respective cylinders according to the respective output fuelinjection quantities #nTout.

The above calculation of the output fuel injection quantities #nTout andthe fuel injection of the internal combustion engine 1 are carried outin successive cycle times synchronous with the crankshaft angle periodof the internal combustion engine 1 for controlling the air-fuel ratioof the internal combustion engine 1 in order to converge the output KACTof the LAF sensor 5 (the detected air-fuel ratio) toward the targetair-fuel ratio KCMD. While the feedback manipulated variable kstr fromthe adaptive controller 18 is being used as the feedback correctioncoefficient KFB, the output KACT of the LAF sensor 5 is quicklyconverged toward the target air-fuel ratio KCMD with high stabilityagainst behavioral changes such as changes in the operating state of theinternal combustion engine 1 or characteristic changes thereof. Aresponse delay of the internal combustion engine 1 is also appropriatelycompensated for.

Concurrent with the above fuel control for the internal combustionengine 1, the exhaust-side control unit 7 a executes a main routineshown in FIG. 13 in control cycles of a constant period.

As shown in FIG. 13, the exhaust-side control unit 7 a decides whetherthe processing of the exhaust-side main processor 13 is to be executedor not, and sets a value of a flag f/prim/cal indicative of whether theprocessing is to be executed or not in STEP1. When the value of the flagf/prim/cal is “0”, it means that the processing of the exhaust-side mainprocessor 13 is not to be executed, and when the value of the flagf/prim/cal is “1”, it means that the processing of the exhaust-side mainprocessor 13 is to be executed.

The deciding subroutine in STEP1 is shown in detail in FIG. 14. As shownin FIG. 14, the exhaust-side control unit 7 a decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEP1-1, STEP1-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the exhaust-side main processor 13 are not accurateenough, the value of the flag f/prism/cal is set to “0” in STEP1-6.Then, in order to initialize the identifier 25 as described later on,the value of a flag f/id/reset indicative of whether the identifier 25is to be initialized or not is set to “1” in STEP1-7. When the value ofthe flag f/id/reset is “1”, it means that the identifier 25 is to beinitialized, and when the value of the flag f/id/reset is “0”, it meansthat the identifier 25 is not to be initialized.

The exhaust-side control unit 7 a decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEP1-3. The exhaust-side control unit 7 a decides whether the ignitiontiming of the internal combustion engine 1 is retarded for earlyactivation of the catalytic converter 3 immediately after the start ofthe internal combustion engine 1 or not in STEP1-4. If the conditions ofthese steps are satisfied, then since the target air-fuel ratio KCMDcalculated to adjust the output VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET is not used for the fuel control for the internalcombustion engine 1, the value of the flag f/id/cal is set to “0” inSTEP1-6, and the value of the flag f/id/reset is set to “1”in order toinitialize the identifier 25 in STEP1-7.

In FIG. 13, after the above deciding subroutine, the exhaust-sidecontrol unit 7 a decides whether a process of identifying (updating) thegain coefficients a1, a1, b1 with the identifier 25 is to be executed ornot, and sets a value of a flag f/id/cal indicative of whether theprocess of identifying (updating) the gain coefficients a1, a1, b1 is tobe executed or not in STEP2. When the value of the flag f/id/cal is “0”,it means that the process of identifying (updating) the gaincoefficients a1, a1, b1 is not to be executed, and when the value of theflag f/id/cal is “1”, it means that the process of identifying(updating) the gain coefficients a1, a1, b1 is to be executed.

In the deciding process of STEP2, the exhaust-side control unit 7 adecides whether the throttle valve of the internal combustion engine 1is substantially fully open or not, and also decides whether the supplyof fuel to the internal combustion engine 1 is being stopped or not. Ifeither one of these conditions is satisfied, then since it is difficultto adjust the gain coefficients a1, a1, b1 appropriately, the value ofthe flag f/id/cal is set to “0”. If neither one of these conditions issatisfied, then the value of the flag f/id/cal is set to “1” to identify(update) the gain coefficients a1, a1, b1 with the identifier 25.

The exhaust-side control unit 7 a determines a present value of thetarget value VO2/TARGET for the output VO2/OUT of the O₂ sensor 6 fromthe present rotational speed NE and intake pressure PB of the internalcombustion engine 1 according to a predetermined map in STEP3.Thereafter, the exhaust-side control unit 7 a calculates the latestdifferential outputs kact(k) (=KACT−FLAF/BASE), VO2(k)(=VO2/OUT−VO2/TARGET) respectively with the subtractors 11, 12 in STEP4.Specifically, the subtractors 11, 12 select latest ones of thetime-series data read and stored in the non-illustrated memory in STEPashown in FIG. 11, and calculate the differential outputs kact(k),VO2(k). For calculating the differential output VO2(k), the exhaust-sidecontrol unit 7 a uses the target value VO2/TARGET determined in STEP3 inthe present control cycle. The differential outputs kact(k), VO2(k), aswell as data given in the past, are stored in a time-series manner in amemory (not shown) in the exhaust-side control unit 7 a.

Then, in STEP5, the exhaust-side control unit 7 a determines the valueof the flag f/prism/cal set in STEP1. If the value of the flagf/prism/cal is “0”, i.e., if the processing of the exhaust-side mainprocessor 13 is not to be executed, then the exhaust-side control unit 7a forcibly sets the SLD manipulating input Usl (the target differentialair-fuel ratio kcmd) to be determined by the sliding mode controller 27,to a predetermined value in STEP14. The predetermined value may be afixed value (e.g., “0”) or the value of the SLD manipulating input Usldetermined in a preceding control cycle.

After the SLD manipulating input Usl is set to the predetermined valuein STEP12, the exhaust-side control unit 7 a adds the reference valueFLAF/BASE to the SLD manipulating input Usl for thereby determining atarget air-fuel ratio KCMD in the present control cycle in STEP 15.Then, the processing in the present control cycle is finished.

If the value of the flag f/prism/cal is “1” in STEP5, i.e., if theprocessing of the exhaust-side main processor 13 is to be executed, thenthe exhaust-side control unit 7 a effects the processing of theidentifier 25 in STEP6.

The processing subroutine of STEP6 is shown in detail in FIG. 15.

The identifier 25 determines the value of the flag f/id/cal set in STEP2in STEP6-1. If the value of the flag f/id/cal is “0”, then since theprocess of identifying the gain coefficients a1, a1, b1 with theidentifier 25 is not carried out, control immediately goes back to themain routine shown in FIG. 13.

If the value of the flag f/id/cal is “1”, then the identifier 25determines the value of the flag f/id/reset set in STEP1 with respect tothe initialization of the identifier 25 in STEP6-2. If the value of theflag f/id/reset is “1”, the identifier 25 is initialized in STEP6-3.When the identifier 25 is initialized, the identified gain coefficientsa1 hat, a2 hat, b1 hat are set to predetermined initial values (theidentified gain coefficient vector Θ according to the equation (4) isinitialized), and the elements of the matrix P (diagonal matrix)according to the equation (9) are set to predetermined initial values.The value of the flag f/id/reset is reset to “0”.

Then, the identifier 25 calculates the identified differential outputVO2(k) hat using the present identified gain coefficients a1(k−1) hat,a2(k−1) hat, b1(k−1) hat and the past data VO2(k−1), VO2(k−2),kact(k−d−1) of the differential outputs VO2, kact calculated in eachcontrol cycle in STEP3, according to the equation (3) in STEP6-4.

The identifier 25 then calculates the vector Kθ(k) to be used indetermining the new identified gain coefficients a1 hat, a2 hat, b1 hataccording to the equation (9) in STEP6-5. Thereafter, the identifier 25calculates the identified error id/e(k), i.e., the difference betweenthe identified differential output VO2 hat and the actual differentialoutput VO2 (see the equation (7)), in STEP6-6.

The identified error id/e obtained in STEP6-6 may basically becalculated according to the equation (7). In the present embodiment,however, a value (=VO2(k)−VO2(k) hat) calculated according to theequation (7) from the differential output VO2 acquired in each controlcycle in STEP4 (see FIG. 13), and the identified differential output VO2hat calculated in each control cycle in STEP6-4 is filtered withlow-pass characteristics to calculate the identified error id/e(k).

This is because since the object exhaust system E including thecatalytic converter 3 generally has low-pass characteristics, it ispreferable to attach importance to the low-frequency behavior of theobject exhaust system E in appropriately identifying the gaincoefficients a1, a2, b1 of the exhaust system model.

Both the differential output VO2 and the identified differential outputVO2 hat may be filtered with the same low-pass characteristics. Forexample, after the differential output VO2 and the identifieddifferential output VO2 hat have separately been filtered, the equation(7) may be calculated to determine the identified error id/e(k). Theabove filtering is carried out by a moving average process which is adigital filtering process.

Thereafter, the identifier 25 calculates a new identified gaincoefficient vector Θ(k), i.e., new identified gain coefficients a1(k)hat, a2(k) hat, b1(k) hat, according to the equation (8) using theidentified error id/e(k) determined in STEP6-6 and Kθ calculated inSETP5-5 in STEP6-7.

After having calculated the new identified gain coefficients a1(k) hat,a2(k) hat, b1(k) hat, the identifier 25 further limits the values of thegain coefficients a1 hat, a2 hat, b1 hat (elements of the identifiedgain coefficient vector Θ) to meet predetermined conditions in STEP6-8.Then, the identifier 25 updates the matrix P(k) according to theequation (10) for the processing of a next control cycle in STEP6-9,after which control returns to the main routine shown in FIG. 13.

The process of limiting the identified gain coefficients a1 hat, a2 hat,b1 hat in STEP6-8 comprises a process of limiting the values of theidentified gain coefficients a1 hat, a2 hat, b1 hat to a certaincombination, i.e., a process of limiting a point (a1 hat, a2 hat) to agiven region on a coordinate plane which has the identified gaincoefficients a1 hat, a2 hat as its component, and a process of limitingthe value of the identified gain coefficient b1 hat within a givenrange. According to the former process, if a point (a1(k) hat, a2(k)hat) on the coordinate plane which is determined by the identified gaincoefficients a1(k) hat, a2(k) hat calculated in STEP6-7 deviates fromthe given region on the coordinate plane, then the values of theidentified gain coefficients a1(k) hat, a2(k) hat are forcibly limitedto the values of the point in the given region. According to the latterprocess, if the value of the identified gain coefficient b1 hatcalculated in STEP6-7 exceeds the upper or lower limit of the givenrange, then the value of the identified gain coefficient b1 hat isforcibly limited to the upper or lower limit of the given range.

The above process of limiting the identified gain coefficients a1 hat,a2 hat, b1 hat serves to maintain stability of the SLD manipulatinginput Usl (the target differential air-fuel ratio kcmd) calculated bythe sliding mode controller 27, and hence the target air-fuel ratioKCMD.

Specific details of the process of limiting the identified gaincoefficients a1 hat, a2 hat, b1 hat are described in Japanese laid-openpatent publication No. 11-153051 and U.S. Pat. No. 6,112,517, and willnot be described herein.

The preceding values a1(k−1) hat, a2(k−1) hat, b1(k−1) hat of theidentified gain coefficients used to determine the new identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat in STEP6-7 are the valuesof the identified gain coefficients after which have been limited inSTEP6-8 in the preceding control cycle.

Details of the processing of the identifier 25 in STEP6 shown in FIG. 13have been described above.

In FIG. 13, after the processing of the identifier 25 has been carriedout, the exhaust-side control unit 7 a determines the values of the gaincoefficients a1, a2, b1 in STEP7. Specifically, if the value of the flagf/id/cal set in STEP2 is “1”, i.e., if the gain coefficients a1, a2, b1have been identified by the identifier 25, then the gain coefficientsa1, a2, b1 are set to the latest identified gain coefficients a1(k) hat,a2(k) hat, b1(k) hat determined by the identifier 25 in STEP6 (limitedin STEP6-8). If the value of the flag f/id/cal is “0”, i.e., if the gaincoefficients a1, a2, b1 have not been identified by the identifier 25,then the gain coefficients a1, a2, b1 are set to predetermined values,respectively.

Then, the exhaust-side control unit 7 a effects a processing operationof the estimator 26, i.e., calculates the estimated differential outputVO2 bar, in STEP8.

The estimator 26 calculates the coefficients α1, α2, βj (j=1, 2, . . . ,d) to be used in the equation (13), using the gain coefficients a1, a2,b1 determined in STEP6 (these values are basically the identified gaincoefficients a1 hat, a2 hat, b1 hat) according to the equation (12).

Then, the estimator 26 calculates the estimated differential outputVO2(k+d) bar (estimated value of the differential output VO2 after thetotal dead time d from the time of the present control cycle) accordingto the equation (13), using the time-series data VO2(k), VO2(k−1), frombefore the present control cycle, of the differential output VO2 of theO₂ sensor calculated in each control cycle in STEP4, the time-seriesdata kact(k−j) (j=0, . . . , d1), from before the present control cycle,of the differential output kact of the LAF sensor 5, the time-seriesdata kcmd(k−j) (=Usl(k−j), j=1, . . . , d2−1), from before the precedingcontrol cycle, of the target differential air-fuel ratio kcmd (=the SLDmanipulating input Usl) given in each control cycle from the slidingmode controller 27, and the coefficients α1, α2, βj calculated asdescribed above.

Then, the exhaust-side control unit 7 a calculates the SLD manipulatinginput Usl (=the target differential air-fuel ratio kcmd) with thesliding mode controller 27 in STEP9.

Specifically, the sliding mode controller 27 calculates a value σ(k+d)bar (corresponding to an estimated value, after the total dead time d,of the linear function σ defined according to the equation (15)), afterthe total dead time d from the present control cycle, of the switchingfunction σ bar defined according to the equation (25), using thetime-series data VO2(k+d) bar, VO2(k+d−1) bar of the estimateddifferential output VO2 bar determined by the estimator 26 in STEP8.

At this time, the sliding mode controller 27 keeps the value of theswitching function σ bar within a predetermined allowable range. If thevalue σ(k+d) bar determined as described above exceeds the upper orlower limit of the allowable range, then the sliding mode controller 27forcibly limits the value σ(k+d) bar to the upper or lower limit of theallowable range. This is because if the value of the switching functionσ bar were excessive, the reaching control law input Urch would beexcessive, and the adaptive control law Uadp would change abruptly,tending to impair the stability of the process of converging the outputVO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET.

Then, the sliding mode controller 27 accumulatively adds values σ((k+d)bar·ΔT, produced by multiplying the value σ((k+d) bar of the switchingfunction σ bar by the period ΔT (constant period) of the control cyclesof the exhaust-side control unit 7 a. That is, the sliding modecontroller 27 adds the product σ((k+d) bar·ΔT of the value σ((k+d) barand the period ΔT calculated in the present control cycle to the sumdetermined in the preceding control cycle, thus calculating anintegrated value σ bar (hereinafter represented by “Σσ bar”) which isthe calculated result of the term Σ(σ bar·T) of the equation (27).

In the present embodiment, the sliding mode controller 27 keeps theintegrated value Σσ bar in a predetermined allowable range. If theintegrated value Σσ bar exceeds the upper or lower limit of theallowable range, then the sliding mode controller 27 forcibly limits theintegrated value Σσ bar to the upper or lower limit of the allowablerange. This is because if the integrated value Σσ bar were excessive,the adaptive control law Uadp determined according to the equation (27)would be excessive, tending to impair the stability of the process ofconverging the output VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET.

Then, the sliding mode controller 27 calculates the equivalent controlinput Ueq, the reaching control law input Urch, and the adaptive controllaw Uadp according to the respective equations (24), (26), (27), usingthe time-series data VO2(k+d)bar, VO2(k+d−1) bar of the present and pastvalues of the estimated differential output VO2 bar determined by theestimator 26 in STEP8, the value σ(k+d) bar of the switching function σand its integrated value Σσ bar which are determined as described above,and the gain coefficients a1, a1, b1 determined in STEP 6 (which arebasically the gain coefficients a1(k) hat, a2(k) hat, b1(k) hat).

The sliding mode controller 27 then adds the equivalent control inputUeq, the reaching control law input Urch, and the adaptive control lawUadp to calculate the SLD manipulating input Usl, i.e., the input (=thetarget differential air-fuel ratio kcmd) to be applied to the objectexhaust system E for converging the output signal VO2/OUT of the O₂sensor 6 toward the target value VO2/TARGET.

After the SLD manipulating input Usl has been calculated, theexhaust-side control unit 7 a determines the stability of the adaptivesliding mode control process carried out by the sliding mode controller27, or more specifically, the ability of the controlled state of theoutput VO2/OUT of the O₂ sensor 6 based on the adaptive sliding modecontrol process (hereinafter referred to as “SLD controlled state”), andsets a value of a flag f/sld/stb indicative of whether the SLDcontrolled state is stable or not in STEP10.

The determining subroutine of STEP10 is shown in detail in FIG. 16.

As shown in FIG. 16, the exhaust-side control unit 7 a calculates adifference Δσ bar (corresponding to a rate of change of the switchingfunction σ bar) between the present value σ(k+d) bar of the switchingfunction σ bar calculated in STEP8 and a preceding value σ(k+d−1) barthereof in STEP10-1.

Then, the exhaust-side control unit 7 a decides whether or not a productΔσ·σ(k+d) bar (corresponding to the time-differentiated function of aLyapunov function σ bar²/2 relative to the σ bar) of the difference Δσbar and the present value σ(k+d) bar is equal to or smaller than apredetermined value ε (≦0) in STEP10-2.

The difference Δσ·σ(k+d) bar (hereinafter referred to as “stabilitydetermining parameter Pstb”) will be described below. If the stabilitydetermining parameter Pstb is greater than 0 (Pstb>0), then the value ofthe switching function σ bar is basically changing away from “0”. If thestability determining parameter Pstb is equal to or smaller than 0(Pstb≦0), then the value of the switching function σ bar is basicallyconverged or converging to “0”. Generally, in order to converge acontrolled variable to its target value according to the sliding modecontrol process, it is necessary that the value of the switchingfunction be stably converged to “0”. Basically, therefore, it ispossible to determine whether the SLD controlled state is stable orunstable depending on whether or not the value of the stabilitydetermining parameter Pstb is equal to or smaller than 0.

If, however, the stability of the SLD controlled state is determined bycomparing the value of the stability determining parameter Pstb with“0”, then the determined result of the stability is affected even byslight noise contained in the value of the switching function σ bar.According to the present embodiment, therefore, the predetermined valueε with which the stability determining parameter Pstb is to be comparedin STEP10-2 is of a positive value slightly greater than “0”.

If Pstb>ε in STEP10-2, then the SLD controlled state is judged as beingunstable, and the value of a timer counter tm (count-down timer) is setto a predetermined initial value T_(M) (the timer counter tm is started)in order to inhibit the determination of the target air-fuel ratio KCMDusing the SLD manipulating input Usl calculated in STEP8 for apredetermined time in STEP10-4. Thereafter, the value of the flagf/sld/stb is set to “0” in STEP10-5, after which control returns to themain routine shown in FIG. 13.

If Pstb≦ε in STEP10-2, then the exhaust-side control unit 7 a decideswhether the present value σ((k+d) bar of the switching function σ barfalls within a predetermined range or not in STEP10-3.

If the present value σ(k+d) bar of the switching function σ bar does notfall within the predetermined range, then since the present valueσ((k+d) bar i spaced widely apart from “0”, the SLD controlled state isconsidered to be unstable. Therefore, if the present value σ((k+d) barof the switching function σ bar does not fall within the predeterminedrange in STEP10-3, then the SLD controlled state is judged as beingunstable, and the processing of STEP10-4 through STEP10-5 is executed tostart the timer counter tm and set the value of the flag f/sld/stb to“0”.

In the present embodiment, since the value of the switching function σbar is limited within the allowable range in STEP9, the decisionprocessing in STEP10-3 may be dispensed with.

If the present value σ(k+d) bar of the switching function σ bar fallswithin the predetermined range in STEP10-3, then the exhaust-sidecontrol unit 7 a counts down the timer counter tm for a predeterminedtime Δtm in STEP10-6. The exhaust-side control unit 7 a then decideswhether or not the value of the timer counter tm is equal to or smallerthan “0”, i.e., whether a time corresponding to the initial value T_(M)has elapsed from the start of the timer counter tm or not, in STEP10-7.

If tm>0, i.e., if the timer counter tm is still measuring time and itsset time has not yet elapsed, then since no substantial time has elapsedafter the SLD controlled state is judged as unstable in STEP10-2 orSTEP10-3, the SLD controlled state tends to become unstable. Therefore,if tm>0 in STEP10-7, then the value of the flag f/sld/stb is set to “0”in STEP10-5.

If tm≦0 in STEP10-7, i.e., if the set time of the timer counter tm haselapsed, then the SLD controlled stage is judged as being stable, andthe value of the flag f/sld/stb is set to “1” in STEP10-8.

According to the above processing, if the SLD controlled state is judgedas being unstable, then the value of the flag f/sld/stb is set to “0”,and if the SLD controlled state is judged as being stable, then thevalue of the flag f/sld/stb is set to “1”.

The above process of determining the stability of the SLD controlledstate is by way of illustrative example only. The stability of the SLDcontrolled state may be determined by any of various other processes.For example, in each given period longer than the control cycle, thefrequency with which the value of the stability determining parameterPstb in the period is greater than the predetermined value ε is counted.If the frequency is in excess of a predetermined value, then the SLDcontrolled state is judged as unstable. Otherwise, the SLD controlledstate is judged as stable.

Referring back to FIG. 13, after a value of the flag f/sld/stbindicative of the stability of the SLD controlled state has been set,the exhaust-side control unit 7 a determines the value of the flagf/sld/stb in STEP11. If the value of the flag f/sld/stb is “1”, i.e., ifthe SLD controlled state is judged as being stable, then the slidingmode controller 27 limits the SLD manipulating input Usl calculated inSTEP 9 in STEP12. Specifically, the sliding mode controller 27determines whether the present value of the SLD manipulating input Uslcalculated in STEP8 falls in a predetermined allowable range or not. Ifthe present value of the SLD manipulating input Usl exceeds the upper orlower limit of the allowable range, then the sliding mode controller 27forcibly limits the present value Usl(k) of the SLD manipulating inputUsl to the upper or lower limit of the allowable range.

The SLD manipulating input Usl (=the target differential air-fuel ratiokcmd) limited in STEP12 is stored in a memory (not shown) in atime-series fashion, and will be used in the processing operation of theestimator 26.

Then, the deteriorated state evaluating means 13 b of the exhaust-sidemain processor 13 performs the process of evaluating the deterioratedstate of the catalytic converter 3 (described later on) in STEP13. Thesliding mode controller 27 adds the reference value FLAF/BASE to the SLDmanipulating input Usl limited in STEP12, thus calculating the targetair-fuel ratio KCMD in STEP15. The processing in the present controlcycle is now put to an end.

If f/sld/stb=0 in STEP11, i.e., if the SLD controlled state is judged asunstable, then the exhaust-side control unit 7 a forcibly sets the SLDmanipulating input Usl in the present control cycle to a predeterminedvalue (the fixed value or the preceding value of the SLD manipulatinginput Usl) in STEP14. The exhaust-side control unit 7 a calculates thetarget air-fuel ratio KCMD according to the equation (28) in STEP 15.Then, the processing in the present control cycle is finished.

The target air-fuel ratio KCMD finally determined in STEP15 is stored ina memory (not shown) in a time-series fashion in each control cycle.When the general feedback controller 15 is to use the target air-fuelratio KCMD determined by the exhaust-side control unit 7 a (see STEPf inFIG. 11), the latest one of the time-series data of the target air-fuelratio KCMD thus stored is selected.

The process of evaluating the deteriorated state of the catalyticconverter 3 in STEP13 will be described below with reference to FIG. 17.

The deteriorated state evaluating means 13 b calculates the value of thedeterioration evaluating linear function σ determined according to theequation (15), from the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6 which is calculated in STEP4shown in FIG. 13, i.e., the present value of the differential output VO2and the past value thereof in the preceding control cycle, in STEP13-1.

The values of the coefficients s1, s2 used to calculate the value of thedeterioration evaluating linear function σ are identical to the valuesof the coefficients s1, s2 used by the sliding mode controller 27 todetermine the value of the switching function σ bar.

Then, the deteriorated state evaluating means 13 b determines the valueof a flag F/DONE in STEP13-2. When the value of the flag F/DONE is “1”,then it indicates that the evaluation of the deteriorated state of thecatalytic converter 3 is completed during the present operation of theinternal combustion engine 1, and when the value of the flag F/DONE is“0”, then it indicates that the evaluation of the deteriorated state ofthe catalytic converter 3 is not completed during the present operationof the internal combustion engine 1. The value of the flag F/DONE is setto “1” in STEP13-5 to be described later on. When the internalcombustion engine 1 starts to operate, the value of the flag F/DONE isinitialized to “0”.

If F/DONE=0, i.e., if the evaluation of the deteriorated state of thecatalytic converter 3 is not completed, then the deteriorated stateevaluating means 13 b performs a process of determining a varying stateof the exhaust gas volume (the rate of flow of the exhaust gas throughthe exhaust pipe 2) in STEP13-3. More specifically, the deterioratedstate evaluating means 13 b determines whether the exhaust gas volume iskept at a substantially constant level, i.e., in a cruise state, or not,and sets the value of a flag F/CRS. When the value of the flag F/CRS is“1”, then it indicates that the exhaust gas volume is in the cruisestate, and when the value of the flag F/CRS is “0”, then it indicatesthat the exhaust gas volume is not in the cruise state. The process ofdetermining a varying state of the exhaust gas volume is carried out ina period of 1 second, for example (hereinafter referred to as “exhaustgas volume variation determining period”) longer than the period (30-100ms) of the control cycles of the exhaust-side control unit 7 a, and isshown in detail in FIG. 18.

As shown in FIG. 18, the deteriorated state evaluating means 13 bcalculates an estimated value ABSV of the present exhaust gas volume(hereinafter referred to as “estimated exhaust gas volume”) from thedetected data of the present rotational speed NE and intake pressure PBof the internal combustion engine 1 according to the following equation(42) in STEP13-3-1: $\begin{matrix}{{ABSV} = {\frac{NE}{1500} \cdot {PB} \cdot {SVPRA}}} & (42)\end{matrix}$

In the present embodiment, the exhaust gas volume when the rotationalspeed of the internal combustion engine 1 is 1500 rpm is used as areference. Therefore, the detected value of the rotational speed NE isdivided by “1500” in the above equation (42). In the equation (42),SVPRA represents a predetermined constant depending on the displacementof the internal combustion engine 1.

Instead of estimating the exhaust gas volume as described above, theexhaust gas volume may be estimated from the fuel supply quantity andintake air quantity of the internal combustion engine 1 or may bedirectly detected using flow sensor.

Then, the deteriorated state evaluating means 13 b effects apredetermined filtering process on the estimated exhaust gas volumecalculated in STEP13-3-1 in each exhaust gas volume variationdetermining period for thereby determining an exhaust gas volumevariation parameter SVMA that represents the varying state of theexhaust gas volume in STEP13-3-2.

The above filtering process is expressed by the following equation (43):

SVMA=(ABSV(n)−ABSV(n−1))+(ABSV(n−2)−ABSV(n−3))+(ABSV(n−4)−ABSV(n−5))  (43)

Specifically, the exhaust gas volume variation parameter SVMA iscalculated by determining a moving average of changes of the estimatedexhaust gas volume ABSV over a plurality of exhaust gas volume variationdetermining periods (three exhaust gas volume variation determiningperiods in the present embodiment). In the equation (43), “n” representsthe ordinal number of the cycle of the exhaust gas volume variationdetermining period.

The exhaust gas volume variation parameter SVMA thus calculatedrepresents a rate of change in the estimated exhaust gas volume ABSV.Consequently, as the value of the exhaust gas volume variation parameterSVMA is closer to “0”, the time-dependent change of the estimatedexhaust gas volume ABSV is smaller, i.e., the estimated exhaust gasvolume ABSV is substantially constant.

Then, the deteriorated state evaluating means 13 b compares the squareof the exhaust gas volume variation parameter SVMA, i.e., the squareSVMA², with a predetermined value δ in STEP13-3-3. The predeterminedvalue δ is a positive value near “0”.

If SVMA²≧δ, i.e., if the present exhaust gas volume suffers a relativelylarge variation, then the deteriorated state evaluating means 13 b setsthe value of a timer counter (count-down timer) TMCRSJUD to apredetermined initial value X/TMCRSJST in STEP13-3-4. As the exhaust gasvolume is not in the cruise state, i.e., the exhaust gas volume is notkept at a substantially constant level, the deteriorated stateevaluating means 13 b sets the flag F/CRS to “0” in STEP13-3-5, afterwhich control returns to the main routine shown in FIG. 17.

If SVMA²<δ in STEP13-3-3, i.e., if the present exhaust gas volumesuffers a relatively small variation, then the deteriorated stateevaluating means 13 b counts down the value of the timer counterTMCRSJUD by a predetermined value in each exhaust gas volume variationdetermining period as long as the present exhaust gas volume suffers arelatively small variation, in STEP13-3-6. Then, the deteriorated stateevaluating means 13 b determines whether or not the value of the timercounter TMCRSJUD becomes “0” or smaller, i.e., whether the set time oftimer counter TMCRSJUD has elapsed or not, in STEP13-3-7.

If TMCRSJUD≦0, i.e., if the set time of the timer counter TMCRSJUD haselapsed, then the deteriorated state evaluating means 13 b decides thatthe exhaust gas volume is in the cruise state, and holds the value ofthe timer counter TMCRSJUD to “0” in STEP13-3-8. Then, the deterioratedstate evaluating means 13 b sets the value of the flag F/CRS to “1” inSTEP13-3-9, after which control returns to the main routine shown inFIG. 17.

If TMCRSJUD>0 in STEP13-3-7, i.e., if the set time of the timer counterTMCRSJUD has not elapsed, then the deteriorated state evaluating means13 b sets the value of the flag F/CRS to “0” in STEP13-3-5, after whichcontrol returns to the main routine shown in FIG. 17.

The processing sequence described above with reference to FIG. 17represents the processing in STEP13-3 shown in FIG. 17. According to theprocessing in STEP13-3, if the square SVMA² of the exhaust gas volumevariation parameter SVMA is SVMA²<δ, i.e., the variation of the exhaustgas volume is small, continuously for a time, e.g., 10 to 15 seconds,corresponding to the initial value X/TMCRSJST of the timer counterTMCRSJUD, the deteriorated state evaluating means 13 b decides that theexhaust gas volume is in the cruise state, and sets the value of theflag F/CRS to “1”. Otherwise, the deteriorated state evaluating means 13b decides that the exhaust gas volume is not in the cruise state, andsets the value of the flag F/CRS to “0”.

The processing in STEP13-3 allows a proper recognition of the state inwhich the exhaust gas volume is maintained at a substantially constantlevel. In each control cycle of the exhaust-side control unit 7 a in oneexhaust gas volume variation determining period, the value of the flagF/CRS is kept constant.

Referring back to FIG. 17, the deteriorated state evaluating means 13 bperforms a process of calculating the deterioration evaluating parameterLSσ² and an average value VO2/OUTAVE of the output VO2/OUT of the O₂sensor 6 in STEP13-4. The process of calculating the deteriorationevaluating parameter LSσ² and an average value VO2/OUTAVE of the outputVO2/OUT of the O₂ sensor 6 will be described below with reference toFIG. 19.

The deteriorated state evaluating means 13 b determines whether certainconditions for calculating the deterioration evaluating parameter LSσ²are satisfied or not in STEP13-4-1. The conditions include the value ofthe flag F/CRS set in STEP13-3 and the value of the flag f/prism/on setby the engine-side control unit 7 b in STEPd shown in FIG. 11.

If F/CRS=1, i.e., if the exhaust gas volume is in the cruise state, thenthe deteriorated state evaluating means 13 b decides that the conditionfor calculating the basic deterioration evaluating parameter LSσ²(hereinafter referred to as “deterioration evaluating condition”) is notsatisfied. Therefore, without calculating the deterioration evaluatingparameter LSσ² and an average value VO2/OUTAVE of the output VO2/OUT ofthe O₂ sensor 6, control goes back to the main routine shown in FIG. 17.

While the exhaust gas volume is in the cruise state, i.e., while exhaustgas volume is maintained at a substantially constant level, the basicdeterioration evaluating parameter LSσ² is not calculated for thefollowing reason: In the cruise state, the output VO2/OUT of the O₂sensor 6 is likely to be held stably to the target value VO2/TARGET, andhence the value of the deterioration evaluating linear function σ isless apt to change even when the deterioration of the catalyticconverter 3 has progressed. In the cruise state, therefore, the value ofthe deterioration evaluating linear function σ does not tend to have atendency depending on the deteriorated state of the catalytic converter3 described above with reference to FIGS. 5 through 7. In the presentembodiment, therefore, the basic deterioration evaluating parameter LSσ²is not calculated in the cruise state, and hence an average valueVO2/OUTAVE of the output VO2/OUT of the O₂ sensor 6 is not calculatedeither.

If f/prism/on=0 in STEP13-4-1, i.e., if the operation mode of theinternal combustion engine 1 is other than the normal operation mode inwhich the fuel supply of the internal combustion engine 1 is controlleddepending on the target air-fuel ratio KCMD that is determined by thesliding mode controller 27 of the exhaust-side control unit 7 a, thenthe deteriorated state evaluating means 13 b also decides that thedeterioration evaluating condition is not satisfied, and does notcalculate the basic deterioration evaluating parameter LSσ² and anaverage value VO2/OUTAVE of the output VO2/OUT of the O₂ sensor 6, andcontrol returns to the main routine shown in FIG. 17. This is becausefor appropriately evaluating the deteriorated state of the catalyticconverter 3 with the basic deterioration evaluating parameter LSσ², itis preferable to determine the basic deterioration evaluating parameterLSσ² using the data of the differential output VO2 of the O₂ sensor 6that is obtained while the air-fuel ratio of the internal combustionengine 1 is being controlled depending on the target air-fuel ratio KCMDgenerated by the sliding mode controller 27 according to the adaptivesliding mode control process.

In STEP13-4-1, the deteriorated state evaluating means 13 b alsodetermines whether the speed of the vehicle with the internal combustionengine 1 mounted thereon is in a predetermined range or not, whether acertain time has elapsed after the startup of the internal combustionengine 1 or not, and whether the catalytic converter 3 has beenactivated or not. If these conditions are not satisfied, then thedeteriorated state evaluating means 13 b determines that thedeterioration evaluating condition is not satisfied. Therefore, withoutcalculating the basic deterioration evaluating parameter LSσ² and anaverage value VO2/OUTAVE of the output VO2/OUT of the O₂ sensor 6,control goes back to the main routine shown in FIG. 17.

If the deterioration evaluating condition is satisfied in STEP13-4-1 (atthis time, F/CRS=0 and f/prism/on=1), then the deteriorated stateevaluating means 13 b calculates the square σ² of the deteriorationevaluating linear function σ determined in each control cycle of theexhaust-side control unit 7 a in STEP13-1 shown in FIG. 17 inSTEP13-4-2.

The deteriorated state evaluating means 13 b calculates a new basicdeterioration evaluating parameter LSσ²(k) from the present value σ²(k)of the square σ², the present value LSσ²(k−1) of the basic deteriorationevaluating parameter LSσ², and the present value BP(k−1) of the gainparameter BP determined by the recursive formula expressed by theequation (30), according to the equation (29) in STEP13-4-3.

After updating the value of the gain parameter BP according to theequation (30) in STEP13-4-4, the deteriorated state evaluating means 13b increments, by “1”, the value of a counter CB1P which counts thenumber of times that the basic deterioration evaluating parameter LSσ²and the gain parameter BP are updated, which number corresponds to thenumber of values of the deterioration evaluating linear function σ usedto determine the basic deterioration evaluating parameter LSσ², inSTEP13-4-5. The value of the counter CB1P is initialized to “0” at thetime of the startup of the internal combustion engine 1.

After the deteriorated state evaluating means 13 b calculates (updates)an average value VO2/OUTAVE of the output VO2/OUT of the O₂ sensor 6 inSTEP13-4-6, control returns to the main routine shown in FIG. 17. Thedeteriorated state evaluating means 13 b calculates an average valueVO2/OUTAVE by accumulatively adding the present value of the outputVO2/OUT of the O₂ sensor 6 each time the processing in STEP13-4-6 iscarried out and dividing the accumulated sum by the number of times thatthe present value of the output VO2/OUT of the O₂ sensor 6 isaccumulatively added. Therefore, the average value VO2/OUTAVE representsan average value of the output VO2/OUT of the O₂ sensor 6 in relation tothe calculation of the basic deterioration evaluating parameter LSσ².

The values of the basic deterioration evaluating parameter LSσ² and thegain parameter BP which are determined respectively in STEP13-4-3 andSTEP13-4-4, and the accumulated sum of the output VO2/OUT of the O₂sensor 6 and the number of times that the present value of the outputVO2/OUT of the O₂ sensor 6 is accumulatively added, which are used tocalculate the average value of the output VO2/OUT of the O₂ sensor 6 inSTEP13-4-6, are stored in a nonvolatile memory such as an EEPROM or thelike (not shown) when the internal combustion engine 1 is shut off, sothat those values will not be lost when the internal combustion engine 1is not operating. When the internal combustion engine 1 operates nexttime, the stored values of the deterioration evaluating parameter LSσ²,the gain parameter BP, the accumulated sum, and the number of times thatthe accumulatively addition is made, are used as their initial values.The initial values of the deterioration evaluating parameter LSσ² andthe gain parameter BP at the time the internal combustion engine 1operates for the first time are “0” and “1”, respectively. The initialvalues of the accumulated sum and the number of times that theaccumulatively addition is made are “0”. The value of the counter CB1Pis initialized to “0” at the time of the startup of the internalcombustion engine 1.

In FIG. 17, after calculating (updating) the values of the basicdeterioration evaluating parameter LSσ² and the average value VO2/OUTAVEas described above, the deteriorated state evaluating means 13 bcorrects the basic deterioration evaluating parameter LSσ² and evaluatesthe deteriorated state of the catalytic converter 3 based on thecorrected deterioration evaluating parameter CLSσ² in STEP13-5. Theprocess of evaluating the deteriorated state of the catalytic converter3 will be described below with reference to FIG. 20.

The deteriorated state evaluating means 13 b determines whether thepresent value BP(k) of the gain parameter BP and the preceding valueBP(k−1) thereof are substantially equal to each other or not, i.e.,whether the gain parameter BP has substantially been converged or not,in STEP12-5-1, and then determines whether or not the value of thecounter CB1P is equal to or greater than a predetermined value CB1CAT,i.e., whether the number of values of the deterioration evaluatinglinear function σ used to determine the deterioration evaluatingparameter LSσ² has reached the predetermined value CB1CAT or not, inSTEP13-5-2.

In the present embodiment, if the data of the deterioration evaluatingparameter LSσ2 and the gain parameter BP from the preceding operationare not held, i.e., if the values thereof are initialized to “0”, aswhen the battery of the vehicle (not shown) is temporarily removedbefore the internal combustion engine 1 is started or as when theinternal combustion engine 1 operates for the first time, then thepredetermined value to be compared with the value of the counter CB1P inSTEP12-5-2 is set to a value greater than if the data of thedeterioration evaluating parameter LSσ² and the gain parameter BP areheld.

If either of the conditions in STEP13-5-1 and STEP13-5-2 is notsatisfied, then the basic deterioration evaluating parameter LSσ²determined in STEP13-4 in the present control cycle is considered to benot sufficiently converged to the central value of the square σ² of thedeterioration evaluating linear function σ. Therefore, the processing inSTEP13-5 is finished without evaluating the deteriorated state of thecatalytic converter 3.

If either of the conditions in STEP13-5-1 and STEP13-5-2 is satisfied,then the basic deterioration evaluating parameter LSσ² determined inSTEP13-4 in the present control cycle is representative of the centralvalue of the square σ² of the deterioration evaluating linear functionσ. The deteriorated state evaluating means 13 b determines a correctivecoefficient KRESS from the present value (calculated in STEP13-4-6) ofthe average value of the output VO2/OUT of the O₂ sensor 6 based on thedata table shown in FIG. 9 in STEP13-5-3. The deteriorated stateevaluating means 13 b multiplies the present value of the basicdeterioration evaluating parameter LSσ² by the corrective coefficientKRESS, thus correcting the basic deterioration evaluating parameter LSσ²into the corrected deterioration evaluating parameter CLSσ² inSTEP13-5-4.

The deteriorated state evaluating means 13 b then compares the correcteddeterioration evaluating parameter CLSσ² with the threshold CATAGELMTshown in FIG. 9 in STEP13-5-5.

If CLSσ²≧CATAGELMT, then the deteriorated state evaluating means 13 bdecides that the deteriorated state of the catalytic converter 3 is inthe deterioration-in-progress state in which it needs to be replacedimmediately or soon. The deteriorated state evaluating means 13 bcontrols the deterioration indicator 29 to indicate the deterioratedstate of the catalytic converter 3 in STEP13-5-6. After setting thevalue of the flag F/DONE to “1”, indicating that the evaluation of thedeteriorated state of the catalytic converter 3 is completed inSTEP13-5-7, the processing in STEP13-5 is finished.

If CLSσ²<CATAGELMT in STEP13-5-5, since the catalytic converter 3 is inthe non-deteriorated state, the deteriorated state evaluating means 13 bdoes not control the deterioration indicator 29, but sets the value ofthe flag F/DONE to “1” in STEP13-5-7. The processing in STEP13-5 is nowfinished.

The above processing represents the process that is carried out by thedeteriorated state evaluating means 13 b in STEP13 shown in FIG. 13.

In the apparatus according to the above embodiment, the target air-fuelratio calculating means 13 a of the exhaust-side main processor 13sequentially determines a target air-fuel ratio for the internalcombustion engine 1, i.e., a target value for the air-fuel ratio of theexhaust gas entering the catalytic converter 3, according to theadaptive sliding mode control process in order to converge (settle) theoutput VO2/OUT from the O₂ sensor 6 downstream of the catalyticconverter 3 to the target value VO2/TARGET which is establisheddepending on the operating state (the rotational speed NE and the intakepressure PB) of the internal combustion engine 1. The target air-fuelratio calculating means 13 a adjusts the amount of fuel injected intothe internal combustion engine 1 in order to converge the output KACT ofthe LAF sensor 5 to the target air-fuel ratio KCMD, for therebyfeedback-controlling the air-fuel ratio of the internal combustionengine 1 at the target air-fuel ratio KCMD. In this manner, the outputsignal VO2/OUT of the O₂ sensor 6 is converged to the target valueVO2/TARGET, and the catalytic converter 3 can maintain its optimumexhaust gas purifying performance.

Concurrent with the above control of the air-fuel ratio of the internalcombustion engine 1, the deteriorated state evaluating means 13 b of theexhaust-side main processor 13 sequentially determines a deteriorationevaluating linear function σ from the time-series data of thedifferential output VO2 of the O₂ sensor 6. The deteriorated stateevaluating means 13 b determines a basic deterioration evaluatingparameter LSσ² as the central value (the central value of the minimumsquare in the present embodiment) of the square σ² of the deteriorationevaluating linear function σ, according to the sequential statisticprocessing algorithm (the algorithm of the method of weighted leastsquares in the present embodiment). The deteriorated state evaluatingmeans 13 b then corrects the basic deterioration evaluating parameterLSσ² depending on the average value VO2/OUTAVE of the output VO2/OUT ofthe O₂ sensor 6, thus determining a corrected deterioration evaluatingparameter CLSσ². The deteriorated state evaluating means 13 b comparesthe corrected deterioration evaluating parameter CLSσ² with thepredetermined threshold CATAGELMT thereby to evaluate the deterioratedstate of the catalytic converter 3.

In this fashion, it is possible to evaluate the deteriorated state ofthe catalytic converter 3 while maintaining the optimum purifyingperformance of the catalytic converter 3. Because the correcteddeterioration evaluating parameter CLSσ² is produced by correcting thebasic deterioration evaluating parameter LSσ² as the central value ofthe square σ² of the deterioration evaluating linear function σ, withthe average value VO2/OUTAVE of the output VO2/OUT of the O₂ sensor 6,or stated otherwise, depending on what average level of the target valueVO2/TARGET for the output VO2/OUT of the O₂ sensor 6 has been controlledabout, its correlation to the deteriorated state of the catalyticconverter 3 is high, independently of the target value VO2/TARGET forcontrolling the output VO2/OUT of the O₂ sensor 6. As a result, thedeteriorated state of the catalytic converter 3 can appropriately beevaluated based on the corrected deterioration evaluating parameterCLSσ².

In the present embodiment, in situations where the exhaust gas volume ismaintained at a substantially constant level, i.e., in the cruise state,i.e., variations of the exhaust gas volume are small and the value ofthe deterioration evaluating linear function σ is unlikely to change,the deterioration evaluating parameter LSσ² and the average valueVO2/OUTAVE of the output VO2/OUT of the O₂ sensor 6 are not calculated.In other situations, the deterioration evaluating parameter LSσ² and theaverage value VO2/OUTAVE are calculated, and the deteriorated state ofthe catalytic converter 3 is evaluated based on the correcteddeterioration evaluating parameter CLSσ² which is determined from thedeterioration evaluating parameter LSσ² and the average valueVO2/OUTAVE. Therefore, the corrected deterioration evaluating parameterCLSσ² representing the deteriorated state of the catalytic converter 3is highly reliable, allowing the deteriorated state of the catalyticconverter 3 to be evaluated accurately.

In the apparatus according to the present embodiment, therefore, thedeteriorated state of the catalytic converter 3 can be evaluated highlyreliably while the desired purifying performance of the catalyticconverter 3 is reliably maintained.

In the above embodiment, the basic deterioration evaluating parameterLSσ² is corrected depending on the average value VO2/OUTAVE of theoutput VO2/OUT of the O² sensor 6. Since the air-fuel ratio of theinternal combustion engine 1 is controlled to converge the outputVO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET, the averagevalue VO2/OUTAVE of the output VO2/OUT of the O² sensor 6 issubstantially equivalent to the average value of the target valueVO2/TARGET. Consequently, the basic deterioration evaluating parameterLSσ² may be corrected depending on the average value of the target valueVO2/TARGET, rather than the average value VO2/OUTAVE of the outputVO2/OUT of the O₂ sensor 6. According to such a modification, inSTEP13-4-6 (see FIG. 19) in the first embodiment, the average value ofthe target value VO2/TARGET, rather than the average value VO2/OUTAVE ofthe output VO2/OUT of the O₂ sensor 6, is determined, and in STEP13-5-3,the corrective coefficient KRESS is determined from the average value ofthe target value VO2/TARGET based on a data table similar to the datatable shown in FIG. 9. Other processing details are identical to thoseof the first embodiment.

An apparatus for controlling the air-fuel ratio of an internalcombustion engine according to a second embodiment of the presentinvention will be described below with reference to FIGS. 21 through 23.The apparatus according to the second embodiment differs from theapparatus according to the first embodiment as to a portion of theprocess of evaluating the deteriorated state of the catalytic converter3. Those parts of the apparatus and the processing sequence thereofaccording to the second embodiment which are identical to those of theapparatus and the processing sequence thereof according to the firstembodiment are shown in identical figures and represented by identicalreference characters, and will not be described in detail below.

In the second embodiment, the deteriorated state evaluating means 13 bof the exhaust-side main processor 13 performs the process of evaluatingthe deteriorated state of the catalytic converter 3 in STEP13 asfollows:

In FIG. 21, after having carried out STEP13-1 through STEP13-4 in thesame manner as with the first embodiment, the deteriorated stateevaluating means 13 b establishes a threshold CATAGELMT2 for determiningthe deteriorated state of the catalytic converter 3 and evaluating thedeteriorated state of the catalytic converter 3 based on the basicdeterioration evaluating parameter LSσ² in STEP13-5′. The subroutine ofSTEP13-5′ is shown in FIG. 22.

According to the subroutine shown in FIG. 22, after having made thedecisions in STEP13-5-1 and STEP13-5-2 in the same manner as with thefirst embodiment, the deteriorated state evaluating means 13 bdetermines the threshold CATAGELMT2 from the present value (latestvalue) of the average value VO2/OUTAVE of the output VO2/OUT of the O₂sensor 6 determined in STEP13-4 (see FIG. 1) based on a data table shownin FIG. 23 in STEP13-5-3′.

The threshold CATAGELMT2 is a threshold to be compared with the basicdeterioration evaluating parameter LSσ² to determine whether thecatalytic converter 3 is in the non-deteriorated state or thedeterioration-in-progress state. In view of the fact that the basicdeterioration evaluating parameter LSσ² is affected by a change in thetarget value VO2/TARGET for the output VO2/OUT of the O₂ sensor 6, thedata table shown in FIG. 23 for setting the threshold CATAGELMT2 isestablished such that the threshold CATAGELMT2 is of a maximum valuewhen the average value VO2/OUTAVE of the output VO2/OUT of the O₂ sensor6 is a value substantially maximizing the sensitivity of the outputVO2/OUT of the O₂ sensor 6, i.e., the level VO2/OUT1 shown in FIG. 2,and is progressively smaller as the difference |VO2/OUTAVE−VO2OUT1|between the average value VO2/OUTAVE and the level VO2/OUT1 is greater.Specifically, as the target value VO2/TARGET is closer to the levelVO2/OUT1 shown in FIG. 2, the output VO2/OUT of the O₂ sensor 6 andhence value of the deterioration evaluating linear function σ are morelikely to vary. The threshold CATAGELMT2 is established so as toincrease and decrease in accordance with such a tendency.

The deteriorated state evaluating means 13 b then compares the basicdeterioration evaluating parameter LSσ² with the threshold CATAGELMT2 inSTEP13-5-5′. If LSσ²≧CATAGELMT, then the deteriorated state evaluatingmeans 13 b decides that the deteriorated state of the catalyticconverter 3 is in the deterioration-in-progress state, and controls thedeterioration indicator 29 to indicate the deteriorated state of thecatalytic converter 3 in STEP13-5-6. After setting the value of the flagF/DONE to “1” in STEP13-5-7, control returns to the main routine shownin FIG. 13. If LSσ²<CATAGELMT in STEP13-5-5′, since the catalyticconverter 3 is in the non-deteriorated state, the deteriorated stateevaluating means 13 b does not control the deterioration indicator 29,but sets the value of the flag F/DONE to “1” in STEP13-5-7, and controlreturns to the main routine shown in FIG. 13.

Other structural and processing details of the apparatus according tothe second embodiment are identical to those of the apparatus accordingto the first embodiment.

In the second embodiment, the basic deterioration evaluating parameterLSσ² is affected by not only the deteriorated state of the catalyticconverter 3, but also a change in the target value VO2/TARGET for theoutput VO2/OUT of the O₂ sensor 6. By establishing the thresholdCATAGELMT2 in accordance with the effect of a change in the target valueVO2/TARGET on the basic deterioration evaluating parameter LSσ², it ispossible to evaluate the deteriorated state of the catalytic converter 3appropriately irrespective of the effect of a change in the target valueVO2/TARGET, as with the first embodiment. According to the secondembodiment, therefore, the deteriorated state of the catalytic converter3 can be evaluated highly reliably while the desired purifyingperformance of the catalytic converter 3 is reliably maintained.

In the second embodiment, the average value VO2/OUTAVE of the outputVO2/OUT of the O₂ sensor 6 is used to establish the thresholdCATAGELMT2. However, the average value of the target value VO2/TARGETfor the output VO2/OUT of the O₂ sensor 6, rather than the average valueVO2/OUTAVE, may be used to establish the threshold CATAGELMT2. Accordingto such a modification, in STEP13-4 shown in FIG. 21, the average valueof the target value VO2/TARGET is determined, and in STEP13-5-3′ shownin FIG. 22, the threshold CATAGELMT2 is determined from the averagevalue of the target value VO2/TARGET based on a data table similar tothe data table shown in FIG. 23.

The present invention is not limited to the first and second embodimentsdescribed above, but may be modified as follows:

In the first and second embodiments, the central value of the minimumsquare σ² of the deterioration evaluating linear function σ is used asthe basic deterioration evaluating parameter LSσ². However, the centralvalue of the minimum square of the absolute value of the deteriorationevaluating linear function σ may be determined as the basicdeterioration evaluating parameter. According to such a modification, inSTEP12-4 shown in FIG. 17 or FIG. 21, the absolute value of thedeterioration evaluating linear function σ is determined instead of thesquare σ² of the deterioration evaluating linear function σ, and |σ²| inthe equation (29) is replaced with the determined absolute value toobtain a basic deterioration evaluating parameter that exhibits the sametendency as the basic deterioration evaluating parameter LSσ² withrespect to the deteriorated state of the catalytic converter 3. Bycomparing the deterioration evaluating parameter which has been producedby correcting the basic deterioration evaluating parameter depending onthe output VO2/OUT of the O₂ sensor 6 or the average value of the targetvalue VO2/TARGET with a predetermined value, or comparing the basicdeterioration evaluating parameter with a threshold establisheddepending on the output VO2/OUT of the O₂ sensor 6 or the average valueof the target value VO2/TARGET, the deteriorated state of the catalyticconverter 3 can be evaluated in the same manner as with the first andembodiments.

Rather than the square σ² of the deterioration evaluating linearfunction σ or the central value of the minimum square of the absolutevalue, the central value of an average value of the square σ² or theabsolute value may be determined as the basic deterioration evaluatingparameter. Alternatively, a variance of the value of the deteriorationevaluating linear function σ, or more accurately a variation withrespect to “0” and an average value of the square σ² of the value of thedeterioration evaluating linear function σ, or a standard deviation (thesquare root of a variance) may be determined as the basic deteriorationevaluating parameter. The basic deterioration evaluating parameter thusdetermined exhibits the same tendency as the basic deteriorationevaluating parameter LSσ² with respect to the deteriorated state of thecatalytic converter 3.

In the first and second embodiments, the deterioration evaluating linearfunction σ is determined according to the equation (15) whose variablecomponents are represented by two time-series data of the differentialoutput VO2 of the O₂ sensor 6. However, the deterioration evaluatinglinear function may be defined by a linear function whose variablecomponents are represented by more time-series data of the differentialoutput VO2. According to such a modification, the switching function ofthe sliding mode control process is preferably defined by a linearfunction where the time-series data of the differential output VO2included in the deterioration evaluating linear function is replacedwith the time-series data of the estimated differential output VO2 bar.

The deterioration evaluating linear function may alternatively bedetermined by an equation similar to the equation (15) where thedifferential outputs VO2(k), VO2(k−1) of the equation (15) are replacedwith the outputs VO2/OUT(k), VO2/OUT(k−1) of the O₂ sensor 6. Accordingto this modification, the central value of the deterioration evaluatinglinear function is basically represented by “(s1+s2)·VO2/TARGET”. If aparameter representing the degree to which the value of thedeterioration evaluating linear function varies with respect to thecentral value (s1+s2)·VO2/TARGET, such as the square of the differencebetween the central value (s1+s2)·VO2/TARGET and the value of thedeterioration evaluating linear function, or the central value of theminimum square of the absolute value, is determined as the basicdeterioration evaluating parameter, then the deteriorated state of thecatalytic converter 3 can be evaluated in the same manner as with thefirst and second embodiments.

Furthermore, a linear function whose variable components are representedby time-series data of the switching function σ bar according to theequation (6), i.e., time-series data of the estimated differentialoutput VO2 bar of the O₂ sensor 6, may be used as the deteriorationevaluating linear function. It is preferable for the purpose ofincreasing the reliability of the evaluated result to use thedeterioration evaluating linear function σ according to the equation(15) which employs the actual differential output VO2 of the O₂ sensor 6as a variable component, rather than the switching function σ bar whichemploys the estimated differential output VO2 bar that is an estimatedvalue after the total dead time d of the differential output VO2 of theO₂ sensor 6, because the deterioration evaluating linear function σbetter reflects the actual state of the catalytic converter 3.

In the first embodiment, the square σ² of the deterioration evaluatinglinear function σ is used to evaluate the deteriorated state of thecatalytic converter 3. However, it is possible to use the product of thevalue of the linear function σ and its rate of change, which representsthe stability determining parameter Pstb used in STEP10 to determine thestability of the SLD controlled state, for evaluating the deterioratedstate of the catalytic converter 3. In such a modification, if avariance of the product, or more generally a value representing thedegree to which the value of the product varies, is determined as thedeterioration evaluating parameter, then it is possible to evaluate thedeteriorated state of the catalytic converter 3 based on thedeterioration evaluating parameter thus determined.

In the first embodiment, the deteriorated state of the catalyticconverter 3 is evaluated as one of the two states, i.e., thedeterioration-in-progress state and the non-deteriorated state. However,if an increased number of thresholds are used for comparison with thecorrected deterioration evaluating parameter CLSσ² or the basicdeterioration evaluating parameter LSσ², then the deteriorated state ofthe catalytic converter 3 may be evaluated as three or more deterioratedstates. In this case, different evaluations may be indicated dependingon those three or more deteriorated states.

In the first and second embodiments, the adaptive sliding mode controlprocess is employed to calculate the target air-fuel ratio KCMD.However, the sliding mode control process which does not use theadaptive control law (adaptive algorithm) may be employed. In thismodification, the target air-fuel ratio KCMD may be determined accordingto an equation that is similar to the equation (28) except that the termof the adaptive control law input Uadp is removed therefrom.

In the first and second embodiments, the effect of the total dead time dis compensated for by the estimator 26 in calculating the targetair-fuel ratio KCMD. If the dead time of the air-fuel ratio manipulatingsystem is negligibly small, then only the dead time d1 of the objectexhaust system E may be compensated for. In this modification, theestimator 26 sequentially determines in each control cycle the estimatedvalue VO2(k+d1) after the dead time d1 of the differential output VO2 ofthe O₂ sensor 6, according to the following equation (44) which issimilar to the equation (12) except that “kcmd” and “d” are replacedrespectively with “kact” and “d1”: $\begin{matrix}\begin{matrix}{{\overset{\_}{VO2}\left( {k + {d1}} \right)} = {{\alpha \quad {1 \cdot {{VO2}(k)}}} + {\alpha \quad {2 \cdot {{VO2}\left( {k - 1} \right)}}} + {\sum\limits_{j = 1}^{d}{\beta \quad {j \cdot {{kact}\left( {k - j} \right)}}}}}} \\{where} \\{{\alpha_{1} = {\text{the~~first-row,~~first-column~~element~~of~~}A^{d1}}},} \\{{{\alpha 2} = {\text{the~~first-row,~~second-column~~element~~of}\quad A^{d1}}},} \\{\beta_{j} = {\text{the~~first-row~~elements~~of~~}{A^{j - 1} \cdot B}}} \\{A = \begin{bmatrix}{a1} & {a2} \\1 & 0\end{bmatrix}} \\{B = \begin{bmatrix}{b1} \\0\end{bmatrix}} \\{B = \begin{bmatrix}{b1} \\0\end{bmatrix}}\end{matrix} & (44)\end{matrix}$

In this modification, the sliding mode controller 27 determines in eachcontrol cycle the equivalent control input Ueq, the reaching control lawinput Urch, and the adaptive control law input Uadp according toequations which are similar to the equations (24)-(27) except that “d”is replaced with “d1”, and adds the equivalent control input Ueq, thereaching control law input Urch, and the adaptive control law input Uadpto determine the target differential air-fuel ratio kcmd for therebydetermining the target air-fuel ratio KCMD which has been compensatedfor the effect of the dead time d1 of the object exhaust system E.

According to the above modification, the processing of the identifier25, the deteriorated state evaluating means 13 b, and the engine-sidecontrol unit 7 b may be the same as the processing thereof in the firstand second embodiments.

If the dead time d1 of the object exhaust system E as well as the deadtime d1 of the air-fuel ratio manipulating system is negligibly small,then the estimator 26 may be dispensed with. In this modification, theprocessing operation of the sliding mode controller 27 and theidentifier 25 may be performed with d=d1.

In the first and second embodiments, the identifier 25 is employed.However, the gain coefficients a1, a2, b1 of the exhaust system modelmay be of predetermined fixed values, or may be set to suitable valuesfrom the rotational speed and intake pressure of the internal combustionengine 1 using a map.

In the first and second embodiments, the sliding mode control process isemployed as the feedback control process for converging the outputVO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET. However, itis possible to employ another feedback control process to evaluate thedeteriorated state of the catalytic converter 3 while converging theoutput VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET.

Although certain preferred embodiments of the present invention havebeen shown and described in detail, it should be understood that variouschanges and modifications may be made therein without departing from thescope of the appended claims.

What is claimed is:
 1. An apparatus for controlling the air-fuel ratioof an internal combustion engine, comprising: an oxygen concentrationsensor disposed downstream of a catalytic converter which is disposed inan exhaust passage of an internal combustion engine; air-fuel ratiomanipulating means for manipulating the air-fuel ratio of an air-fuelmixture to be combusted in said internal combustion engine to convergean output of said oxygen concentration sensor to a target valueestablished depending on an operating state of the internal combustionengine; parameter generating means for generating a deteriorationevaluating parameter for evaluating a deteriorated state of thecatalytic converter according to an algorithm predetermined from data ofthe output of said oxygen concentration sensor while said air-fuel ratiomanipulating means is manipulating the air-fuel ratio of the air-fuelmixture; and deteriorated state evaluating means for correcting saiddeterioration evaluating parameter depending on an average value of saidtarget value or the output of said oxygen concentration sensor, andevaluating the deteriorated state of said catalytic converter based onthe corrected deterioration evaluating parameter.
 2. An apparatusaccording to claim 1, wherein said deterioration evaluating parametercomprises data representing a variation of a deterioration evaluatinglinear function whose variable components are represented by time-seriesdata of the output of said oxygen concentration sensor.
 3. An apparatusaccording to claim 2, wherein said parameter generating means comprisesmeans for generating said deterioration evaluating parameter byeffecting low-pass filtering on the square value or absolute value ofthe difference between values of the time-series data of the output ofsaid oxygen concentration sensor and a predetermined value as a centralvalue of said deterioration evaluating linear function.
 4. An apparatusaccording to claim 2, wherein said air-fuel ratio manipulating meanscomprises means for sequentially generating a manipulated variable formanipulating said air-fuel ratio according to a sliding mode controlprocess to converge the output of said oxygen concentration sensor tosaid target value, and manipulating said air-fuel ratio depending onsaid manipulated variable, said deterioration evaluating linear functioncomprising a linear function determined depending on a switchingfunction used in said sliding mode control process.
 5. An apparatus forcontrolling the air-fuel ratio of an internal combustion engine,comprising: an oxygen concentration sensor disposed downstream of acatalytic converter which is disposed in an exhaust passage of aninternal combustion engine; air-fuel ratio manipulating means formanipulating the air-fuel ratio of an air-fuel mixture to be combustedin said internal combustion engine to converge an output of said oxygenconcentration sensor to a target value established depending on anoperating state of the internal combustion engine; parameter generatingmeans for generating a deterioration evaluating parameter for evaluatinga deteriorated state of the catalytic converter according to analgorithm predetermined from data of the output of said oxygenconcentration sensor while said air-fuel ratio manipulating means ismanipulating the air-fuel ratio of the air-fuel mixture; anddeteriorated state evaluating means for evaluating the deterioratedstate of said catalytic converter by comparing the deteriorationevaluating parameter with a predetermined decision value; saiddeteriorated state evaluating means comprising: means for establishingsaid decision value depending on an average value of said target valueor the output of said oxygen concentration sensor; and means forevaluating the deteriorated state of said catalytic converter bycomparing the established decision value with said deteriorationevaluating parameter.
 6. An apparatus according to claim 5, wherein saiddeterioration evaluating parameter comprises data representing avariation of a deterioration evaluating linear function whose variablecomponents are represented by time-series data of the output of saidoxygen concentration sensor.
 7. An apparatus according to claim 6,wherein said parameter generating means comprises means for generatingsaid deterioration evaluating parameter by effecting low-pass filteringon the square value or absolute value of the difference between valuesof the time-series data of the output of said oxygen concentrationsensor and a predetermined value as a central value of saiddeterioration evaluating linear function.
 8. An apparatus according toclaim 6, wherein said air-fuel ratio manipulating means comprises meansfor sequentially generating a manipulated variable for manipulating saidair-fuel ratio according to a sliding mode control process to convergethe output of said oxygen concentration sensor to said target value, andmanipulating said air-fuel ratio depending on said manipulated variable,said deterioration evaluating linear function comprising a linearfunction determined depending on a switching function used in saidsliding mode control process.